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Similar Question 1

<p>To test the resistance of a new product to temperature changes, the A product is placed in a controlled environment. The temperature in this environment, as a function of time, can be described by a sine function. The maximum temperature is <code class='latex inline'>120^{o}</code>C, the minimum temperature is <code class='latex inline'>260^{o}</code>C, and the temperature at <code class='latex inline'>t = 0</code> is <code class='latex inline'>30^{o}</code>C. It takes 12 h for the temperature to change from the maximum to the minimum. If the temperature is initially increasing, what is the equation of the sine function that describes the temperature in this environment?</p>

Similar Question 2

<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>a) Determine the vertical translation of the function.</p><p>b) What is the desired period of the function?</p>

Similar Question 3

<p>Determine an equation in the form <code class='latex inline'>y=a\sin kx</code> or <code class='latex inline'>y=a\cos kx</code> for each graph. </p><p><strong>(a)</strong></p><img src="/qimages/525" /><p><strong>(b)</strong></p><img src="/qimages/526" />

Similar Questions

Learning Path

L1
Quick Intro to Factoring Trinomial with Leading a

L2
Introduction to Factoring ax^2+bx+c

L3
Factoring ax^2+bx+c, ex1

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<p>Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At <code class='latex inline'>t = 0</code>, the sparkler is at its highest point above the ground.</p>
<ul>
<li>If no horizontal translations are required to model this situation,
should a sine or cosine function be used?</li>
</ul>

<p>The piston in the engine of a small aircraft moves horizontally relative to the crankshaft, from a minimum distance of 25 cm to a maximum distance of 75 cm. During normal cruise power settings, the piston completes 2100 rpm (revolutions per minute). </p><p>Model the horizontal position, <code class='latex inline'>h</code>, in centimetres, of the piston as a function of time, <code class='latex inline'>t</code>, in seconds.</p>

<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'>
\displaystyle
y = \frac{1}{5}\sin 4(x + \frac{\pi}{2}) - 3
</code></p>

<p>Sketch a graph of each function below on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle
\begin{array}{ccccccc}
&(a) & y = \sin x + 4 & &(b) & y = \sin x -2 \\
&(c) & y = \cos x -5 & &(b) & y = \cos x + 1 \\
\end{array}
</code></p>

<p>Each person's blood pressure is different, but there is a range of blood
pressure values that is considered healthy. The function <code class='latex inline'>P(t) = -20\cos{\displaystyle{\frac{5\pi}{3}}}t+100</code> models the blood pressure, <code class='latex inline'>p</code>, in
millimetres of mercury, at time <code class='latex inline'>t</code>, in seconds, of a person at rest.</p><p>d) What is the range of the function? Explain the meaning of the
range in terms of a person's blood pressure.</p>

<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle
y = \cos x -5
</code></p>

<p>State each as a cosine graph</p><img src="/qimages/285" />

<p>State each as a cosine graph</p><img src="/qimages/284" />

<p>Determine an equation for each cosine function. </p><img src="/qimages/530" />

<p>A wheel with a radius of 1.4 m catches a stone in its tire as it rolls along the ground. The wheel rotates 5 times every second.</p><p>a) draw a graph to show the height of the stone above the ground as a function of the distance rolled for two cycles. The cycle begins when the tire catches the stone.</p><p>b) find an equation of the graph in part (a).</p><p>c) find the height of the stone when the wheel has rolled 10.3 m.</p><p>d) find the distance the wheel has travelled when the stone is 2 m above the ground?</p>

<p>Each person's blood pressure is different, but there is a range of blood
pressure values that is considered healthy. The function <code class='latex inline'>P(t) = -20\cos{\displaystyle{\frac{5\pi}{3}}}t+100</code> models the blood pressure, <code class='latex inline'>p</code>, in
millimetres of mercury, at time <code class='latex inline'>t</code>, in seconds, of a person at rest.</p><p>b) How many times does this person's heart beat each minute?</p>

<p>Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At <code class='latex inline'>t = 0</code>, the sparkler is at its highest point above the ground.</p>
<ul>
<li>What does the period of the sinusoidal function represent in this
situation?</li>
</ul>

<p>State the period, amplitude, horizontal translation, and equation of
the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y = 0.5\cos(4x)</code></p>

<p> Find the information that is listed below for each sine function then sketch it.</p>
<ul>
<li>Amplitude</li>
<li>Any reflections</li>
<li>Period</li>
<li>Phase Shift</li>
<li>Vertical Shift</li>
</ul>
<p><code class='latex inline'>y = 5\sin(x - \frac{\pi}{6}) + 8</code></p>

<p>For each of the following equations, state the parent function and the transformations that were applied. Graph the transformed function.</p><p><code class='latex inline'>\displaystyle
f(x) = -2\sin(3x) + 1, 0 \leq x \leq 360^o
</code></p>

<p>A clock is hanging on a wall, with the centre of the clock 3 m above the floor. Both the minute hand and the second hand are 15 cm long. The hour hand is 8 cm long. For each hand, determine the equation of the cosine function that describes the distance of the tip of the hand above the floor as a function of time. Assume that the time, t, is in minutes and that the distance, <code class='latex inline'>D(t)</code>, is in centimetres. Also assume that <code class='latex inline'>t = 0</code> is midnight.</p>

<p>What is the amplitude? </p><img src="/qimages/980" />

<p>Write an equation for</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{3\pi}{4}</code></p>

<p>The graph shows the distance from a light pole to a car racing around a
circular track. The track is located north of the light pole.</p><img src="/qimages/2171" /><p>d) Determine the time that the car takes to complete one lap of the track.</p>

<p>Sketch a graph of in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{\pi}{3}</code></p>

<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p>
<ul>
<li>Determine the equation of the model in the form <code class='latex inline'>y = a\sin kx</code>.</li>
</ul>

<p>Determine the amplitude, the period, the phase shift, and the vertical translation for each function with respect to <code class='latex inline'>y=\cos x</code>. Then, sketch a graph of the function for two cycles. </p><p><strong>(a)</strong> <code class='latex inline'>\displaystyle{y=3\cos\left(x-\frac{\pi}{4}\right)+6}</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{y=-5\cos\left[\frac{1}{4}\left(x+\frac{4\pi}{3}\right)\right]-5}</code></p><p><strong>(c)</strong> <code class='latex inline'>\displaystyle{7\cos[3\pi(x-2)]}+7</code></p>

<p>Graph <code class='latex inline'>f(x) = \frac{1}{2}\cos(x - \frac{\pi}{4})+3</code> showing:</p>
<ul>
<li>symmetry, if any</li>
<li>x intercept, if any</li>
<li>y intercept, if any</li>
<li>asymptotes, if any</li>
<li>domain </li>
</ul>

<p>At one time, Maple Leaf Village (which no longer exists) had North America?s largest Ferris wheel. The Ferris wheel had a diameter of 56 m, and one revolution took 2.5 min to complete. Riders could see Niagara Falls if they were higher than 50 m above the ground. Sketch three cycles of a graph that represents the height of a rider above the ground, as a function of time, if the rider gets on at a height of 0.5 m at <code class='latex inline'>t = 0</code> min. Then determine the time intervals when the rider could see Niagara Falls.</p>

<p>Sketch a graph of </p><p>cosine function with an amplitude of 5.</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>

<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>cosine function with a phase shift of <code class='latex inline'>\frac{3\pi}{2}</code></li>
</ul>

<p>State the period, amplitude, and equation of the axis of the
trigonometric function that produces each of the following tables
of values. Then use this information to write the equation of the
function.</p><img src="/qimages/497" />

<p>Determine a sinusodial equation for each of the following graphs.</p><img src="/qimages/500" />

<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p>
<ul>
<li>Write the caution of the function in the form <code class='latex inline'>y = \sin[k(x - d)]</code>.</li>
</ul>

<p> Sketch each function and determine its period.</p><p><code class='latex inline'>
\displaystyle
y = |\sin x|
</code></p><p><a href="https://youtu.be/cldcTIigYk4">HINT</a></p>

<p>A pendulum swings back and forth 10 times in 8 s. It swings through
a total horizontal distance of 40 cm. </p><p>c) Write the equation that models this situation.</p>

<p>Identify the key characteristics of <code class='latex inline'>y = -2\cos(4x+\pi)+4</code>, and sketch its graph. Check your graph with a graphing calculator.</p>

<p>Sketch a graph of </p><p>sine function with an amplitude of 3</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>

<p> Sketch each function and determine its period.</p><p><code class='latex inline'>
\displaystyle
y = | 2\sin x + 1|
</code></p>

<p>A cosine curve has an amplitude of 3 units and a period of <code class='latex inline'>3\pi</code> radians.
The equation of the axis is <code class='latex inline'>y = 2</code>, and a horizontal shift of <code class='latex inline'>\frac{\pi}{4}</code> radians
to the left has been applied. </p><p>Sketch a graph of the function in described above. Use your graph to estimate the x-value(s) in the domain <code class='latex inline'>0 < x < 2\pi</code>, where <code class='latex inline'>y = 2.5</code>, to one decimal place.</p>

<p>The graph shows the distance from a light pole to a car racing around a
circular track. The track is located north of the light pole.</p><img src="/qimages/2171" /><p>b) Determine the distance from the light pole to the centre of the track.</p>

<p>Determine the amplitude and the period of each sinusoidal function. Then, transform the graph of <code class='latex inline'>y=\sin x</code> to sketch a graph of each function. </p><p><strong>(i)</strong> <code class='latex inline'>\displaystyle{y=\frac{1}{2}\cos\frac{1}{4}x}</code></p><p><strong>(ii)</strong> <code class='latex inline'>y=-1.5\sin0.2\pi x</code></p><p><strong>(iii)</strong> <code class='latex inline'>y=0.75\cos0.8\pi x</code></p>

<p> Find the information that is listed below for each sine function then sketch it.</p>
<ul>
<li>Amplitude</li>
<li>Any reflections</li>
<li>Period</li>
<li>Phase Shift</li>
<li>Vertical Shift</li>
</ul>
<p><code class='latex inline'>y = -2 \sin\frac{1}{4}(2x + \frac{\pi}{6}) + 1 </code></p>

<p>Consider the function <code class='latex inline'>y=3\cos[\pi(x+2)]-1</code> </p><p><strong>(a)</strong> What is the amplitude?</p><p><strong>(b)</strong> What is the period?</p><p><strong>(c)</strong> Describe the phase shift.</p><p><strong>(d)</strong> Describe the vertical translation.</p>

<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle
y = \sin x -2
</code></p>

<p>Determine an equation for each cosine function. </p><p><strong>(a)</strong></p><img src="/qimages/531" /><p><strong>(b)</strong></p><img src="/qimages/532" />

<p>For the following equation, what is the parent function and the transformation that are applied? </p><p><code class='latex inline'>
y= \sin(3x) + 1
</code></p>

<p>Write an equation for</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{\pi}{3}</code></p>

<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing
technology.</p><p><code class='latex inline'>y = \displaystyle{\frac{1}{2}}\cos\left(\displaystyle{\frac{x}{2}}-\displaystyle{\frac{\pi}{12}}\right)-3</code></p>

<p>The number of hours of daylight in Vancouver can be modelled by a sinusoidal function of time, in days. The longest day of the year is June 21, with 15.7 h of daylight. The shortest day of the year is December 21, with 8.3 h of daylight.</p>
<ul>
<li>Find an equation for n(t), the number of hours of daylight on the nth day of the year.</li>
</ul>

<p> A ferris wheel has a radius of 10 m, and the bottom of the wheel passes 1 m above the ground. If the ferris wheel makes one complete revolution every 20 s, find an equation that gives the height above the ground of a person on the ferris wheel as a function of time. Assume the person gets on the bottom of the ferries wheel.</p>

<p>A sine function is transformed such that it has a single <code class='latex inline'>x</code>-intercept in the interval <code class='latex inline'>[0, \pi]</code>, a period of <code class='latex inline'>\pi</code>, and a <code class='latex inline'>y</code>-intercept of 3. </p><p> Determine an equation for a function that satisfies the properties given.</p>

<p>A pendulum swings back and forth 10 times in 8 s. It swings through
a total horizontal distance of 40 cm. </p><p>b) Describe the transformations necessary to transform <code class='latex inline'>y = \sin{x}</code> into the function you graphed in part a).</p>

<p> Sketch the following graph.</p><p><code class='latex inline'>
\displaystyle
y = 2\sin(2x )
</code></p>

<p>To test the resistance of a new product to temperature changes, the A product is placed in a controlled environment. The temperature in this environment, as a function of time, can be described by a sine function. The maximum temperature is <code class='latex inline'>120^{o}</code>C, the minimum temperature is <code class='latex inline'>260^{o}</code>C, and the temperature at <code class='latex inline'>t = 0</code> is <code class='latex inline'>30^{o}</code>C. It takes 12 h for the temperature to change from the maximum to the minimum. If the temperature is initially increasing, what is the equation of the sine function that describes the temperature in this environment?</p>

<p>Determine the amplitude and the period of each sinusoidal function. Then, transform the graph of <code class='latex inline'>y=\sin x</code> to sketch a graph of each function. </p><p><strong>(a)</strong> <code class='latex inline'>y=5\sin3x</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{y=-3\cos\frac{4}{3}x}</code></p><p><strong>(c)</strong> <code class='latex inline'>y=3\sin\pi x</code></p>

<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following
sets of transformations. For each set of transformations, determine the
equation of the resulting function and sketch its graph.</p><p>horizontal compression by a factor of <code class='latex inline'>\displaystyle{\frac{1}{2}}</code>, horizontal translation <code class='latex inline'>\displaystyle{\frac{\pi}{2}}</code> to the left</p>

<p>Sketch a graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>sine function with a phase shift of <code class='latex inline'>-\frac{3\pi}{4}</code></li>
</ul>

<p> Find the equation of cosine graph with its first maximum at <code class='latex inline'>x = \frac{2\pi}{3}</code> and first minimum at <code class='latex inline'> x = 3\pi</code>. The maximum value of the cosine curve is 10 and the minimum is 2.</p>

<p>Determine a sinusodial equation for each of the following graphs.</p><p> <img src="/qimages/498" /></p>

<p>Write an equation for </p><p>cosine function with an amplitude of 5.</p>

<p> Find the information that is listed below for each sine function then sketch it.</p>
<ul>
<li>Amplitude</li>
<li>Any reflections</li>
<li>Period</li>
<li>Phase Shift</li>
<li>Vertical Shift</li>
</ul>
<p><code class='latex inline'>y = \sin (4x - 6\pi)</code></p>

<p>The graph shows the distance from a light pole to a car racing around a
circular track. The track is located north of the light pole.</p><img src="/qimages/2171" /><p>c) Determine the radius of the track.</p>

<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle
y = \cos x + 1
</code></p>

<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'>
\displaystyle
y = \frac{1}{3} \cos(3x + \frac{\pi}{2}) + 1
</code></p>

<p>Sketch a graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>cosine function with a phase shift of <code class='latex inline'>\frac{5\pi}{6}</code></li>
</ul>

<p>Determine the amplitude, the period, the phase shift, and the vertical translation for each function with respect to <code class='latex inline'>y=\sin x</code>. Then, sketch a graph of the function for two cycles. </p><p><strong>(a)</strong> <code class='latex inline'>\displaystyle{y=3\sin\left(x+\frac{\pi}{4}\right)-1}</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{-2\sin\left[\frac{1}{2}\left(x-\frac{5\pi}{6}\right)\right]+4}</code></p><p><strong>(c)</strong> <code class='latex inline'>y=2\sin[2\pi(x+3)]-2</code></p>

<p>Write an equation for each function.</p>
<ul>
<li>sine function with a phase shift of <code class='latex inline'>\frac{\pi}{2}</code></li>
</ul>

<p>A rung on a hamster wheel, with a radius of 25 cm, is travelling at a
constant speed. It makes one complete revolution in 3 s. The axle of
the hamster wheel is 27 cm above the ground.</p><p>a) Sketch a graph of the height of the rung above the ground during
two complete revolutions, beginning when the rung is closest to
the ground.</p>

<p>Write an equation for each function.</p>
<ul>
<li>cosine function with a phase shift of <code class='latex inline'>\frac{3\pi}{2}</code></li>
</ul>

<p>Each person's blood pressure is different, but there is a range of blood
pressure values that is considered healthy. The function <code class='latex inline'>P(t) = -20\cos{\displaystyle{\frac{5\pi}{3}}}t+100</code> models the blood pressure, <code class='latex inline'>p</code>, in
millimetres of mercury, at time <code class='latex inline'>t</code>, in seconds, of a person at rest.</p><p>a) What is the period of the function? What does the period
represent for an individual?</p>

<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'>
\displaystyle
y = \frac{1}{5} \sin(3x + \frac{\pi}{4}) + 10
</code></p>

<p>State each as a cosine graph</p><img src="/qimages/286" />

<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p>
<ul>
<li>What is the period of the model?</li>
</ul>

<p>The frequency of a periodic function is defined as the number of cycles complete in 1 s and is typically measured in hertz (Hz). It is the reciprocal of the period of a periodic function.</p>
<ul>
<li>One of the A-notes from a flute vibrates <code class='latex inline'>440</code> times in 1 s. It is said to have a frequency of <code class='latex inline'>440</code> Hz. What is the period of the A-note?</li>
</ul>

<p>Determine an equation in the form <code class='latex inline'>y=a\sin kx</code> or <code class='latex inline'>y=a\cos kx</code> for each graph. </p><p><strong>(a)</strong></p><img src="/qimages/525" /><p><strong>(b)</strong></p><img src="/qimages/526" />

<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'>
\displaystyle
y = -\frac{1}{2}\cos 4x - 3
</code></p>

<p>Write an equation for </p><p>cosine function with an amplitude of 2, reflected in the x-axis.</p>

<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing
technology.</p><p><code class='latex inline'>y = 0.5\sin\left(\displaystyle{\frac{x}{4}}-\displaystyle{\frac{\pi}{16}}\right)-5</code></p>

<p> Graph <code class='latex inline'>y = (\sin x)^2, -2\pi \leq x \leq 2\pi.</code></p><p>(a) determine the period;</p><p>(b) graph the function for one complete period.</p>

<p> Given the function <code class='latex inline'>y = \sin 2\pi x</code></p><p>(a) determine the period.</p><p>(b) graph the function for one complete period.</p>

<p>Find an expression that describes the location of each of the following values for <code class='latex inline'>y = cos \theta</code>, where <code class='latex inline'>n \in \mathbb{I}</code> and <code class='latex inline'>\theta</code> is in radians.\</p><p>(a) <code class='latex inline'>\theta-intercepts</code></p><p>(b) maximum values</p><p>(c) minimum values</p>

<p>The frequency of a periodic function is defined as the number of cycles complete in 1 s and is typically measured in hertz (Hz). It is the reciprocal of the period of a periodic function.</p>
<ul>
<li>The sound can be modelled using a sine function of the form <code class='latex inline'>y = \sin kx</code>. What is the value of <code class='latex inline'>k</code>.</li>
</ul>

<p>The Round—Up is a popular ride at many theme parks. It consists of a large wheel that starts in a horizontal position. As the ride picks up rotational speed, the wheel tilts up to an angle of about <code class='latex inline'>\displaystyle \frac{\pi}{4}</code></p><img src="/qimages/2166" /><p> Determine a model for the Round-Up using a cosine function.</p>

<p>A crank 20 cm long, starting at <code class='latex inline'>60^{\circ}</code> above from the horizontal, sweeps out a positive angle at the uniform rate of 10 rad/s.</p><p>(b) How far is the end of the crank from the horizontal line after 0.1 s?</p>

<p>A buoy bobs up and down in the lake. The distance between the highest and lowest points is 1.5m. It takes 6s for the buoy to move from its highest point to its lowest point and back to its highest point. </p><p>Model the vertical displacement, <code class='latex inline'>v</code>, in metres, of the buoy as a function of time, <code class='latex inline'>t</code>, in seconds. Assume that the buoy is at its equilibrium point at <code class='latex inline'>t=0</code>s and that the buoy is on its way down at that time.</p>

<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>cosine function with a phase shift of <code class='latex inline'>\pi</code></li>
</ul>

<p>A popular attraction in many water them parks is the wave pool. Suppose that the wave generator moves back and forth 20 complete cycles every minute to produce waves measuring 1.2 m from the top of the crest to the bottom of the trough.</p>
<ul>
<li>If the operator increases the speed of the wave generator to 30 complete cycles every minute, what change or changes would you expect to see in the waves? Hw would your model change? Justify your answer.</li>
</ul>

<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>Determine the amplitude of the function </p>

<p>A guy wire supporting a telephone pole is secured to the ground at a point 16.7 m from the base of the pole. The wire makes an angle of 48<code class='latex inline'>^{\circ}</code> with the ground.</p><p>Use the equation made from using a reciprocal trigonometric ratio to find the length of the wire to the nearest centimetre. </p>

<p> Sketch the following graph.</p><p><code class='latex inline'>
\displaystyle
y = -3\sin( \frac{5}{2}x )
</code></p>

<p>Graph <code class='latex inline'>f(x) = 3\sin(2(x+1))-1</code> showing:</p>
<ul>
<li>State the transformation mapping</li>
</ul>

<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p>
<ul>
<li>What is the amplitude of the model?</li>
</ul>

<p>Each person's blood pressure is different, but there is a range of blood
pressure values that is considered healthy. The function <code class='latex inline'>P(t) = -20\cos{\displaystyle{\frac{5\pi}{3}}}t+100</code> models the blood pressure, <code class='latex inline'>p</code>, in
millimetres of mercury, at time <code class='latex inline'>t</code>, in seconds, of a person at rest.</p><p>c) Sketch the graph of <code class='latex inline'>y = P(t)</code> for <code class='latex inline'>0 \leq t \leq 6</code>.</p>

<p>A function is said to be even if <code class='latex inline'>f(-x) = f(x)</code> for all values of . A function is said to be odd if <code class='latex inline'>f(-x) = -f(x)</code> for all values of <code class='latex inline'>x</code>.</p><p>Is the function <code class='latex inline'>y = \sin x</code> even, odd, or neither?</p>

<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following
sets of transformations. For each set of transformations, determine the
equation of the resulting function and sketch its graph.</p><p>vertical stretch by a factor of 3, horizontal translation <code class='latex inline'>\displaystyle{\frac{\pi}{2}}</code> to the
right</p>

<p>A popular attraction in many water them parks is the wave pool. Suppose that the wave generator moves back and forth 20 complete cycles every minute to produce waves measuring 1.2 m from the top of the crest to the bottom of the trough.</p>
<ul>
<li>Sketch a graph of the model over two cycles.</li>
</ul>

<p>The number of hours of daylight in Vancouver can be modelled by a sinusoidal function of time, in days. The longest day of the year is June 21, with 15.7 h of daylight. The shortest day of the year is December 21, with 8.3 h of daylight.</p><p>(b) Use your equation to predict the number of hours of daylight in Vancouver on January 30th.</p>

<p>Determine the value of the function <code class='latex inline'>
\displaystyle
y=3\cos\left(\dfrac{2}{3}\left(x + \dfrac{\pi}{4}\right)\right) + 2
</code>
if <code class='latex inline'>x = \frac{\pi}{2}, \frac{3\pi}{4}</code>, and <code class='latex inline'>\frac{11\pi}{6}</code>.</p>

<p>A cosine curve has an amplitude of 3 units and a period of <code class='latex inline'>3\pi</code> radians.
The equation of the axis is <code class='latex inline'>y = 2</code>, and a horizontal shift of <code class='latex inline'>\frac{\pi}{4}</code> radians
to the left has been applied. Write the equation of this function.</p>

<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'>
\displaystyle
y = 20 \sin 3x + 10
</code></p>

<p> Graph each of the following function for one complete period.</p><p><code class='latex inline'>
\displaystyle
\begin{array}{cccc}
&(a) & y = \sin \frac{\pi}{2}x &(b) & y = \sin 4\pi x \\
\end{array}
</code></p>

<p>State each as a cosine graph</p><img src="/qimages/283" />

<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing
technology.</p><p><code class='latex inline'>y = 5\cos\left(x+\displaystyle{\frac{\pi}{4}}\right)-2</code></p>

<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p>
<ul>
<li>Determine the period of the function.</li>
</ul>

<p>Sketch the following graphs.</p><p><code class='latex inline'>
\displaystyle
\begin{array}{cccc}
&(a) & y = -2\sin x &(b)& y = 3 \sin(x ) &(c) & y = -10 \sin( x ) \\
\end{array}
</code></p>

<p>A function is said to be even if <code class='latex inline'>f(-x) = f(x)</code> for all values of . A function is said to be odd if <code class='latex inline'>f(-x) = -f(x)</code> for all values of <code class='latex inline'>x</code>.</p><p>Is the function <code class='latex inline'>y = \cos x</code> even, odd, or neither?</p>

<p>A cosine function has a maximum value of <code class='latex inline'>7</code> and minimum value of <code class='latex inline'>-3</code>.</p>
<ul>
<li>Determine the amplitude of the function.</li>
</ul>

<p>Consider the function <code class='latex inline'>\displaystyle{y=4\sin\left[3\left(x+\frac{\pi}{3}\right)\right]}</code> </p><p><strong>(a)</strong> What is the amplitude?</p><p><strong>(b)</strong> What is the period?</p><p><strong>(c)</strong> Describe the phase shift.</p><p><strong>(d)</strong> Describe the vertical translation.</p>

<p> Answer the following for each graph.</p>
<ul>
<li>amplitude</li>
<li>period</li>
<li>Maximum and Minimum value</li>
</ul>
<p><code class='latex inline'>
\displaystyle
\begin{array}{ccccc}
&(a) & y = 5\cos 3x &(b)& y = -3 \cos \frac{7}{2}x
&(c) &y = -2 \cos(2x + \frac{\pi}{12}) \\
\end{array}
</code></p>

<p>State the period, amplitude, and equation of the axis of the
trigonometric function that produces each of the following tables
of values. Then use this information to write the equation of the
function.</p><p> <img src="/qimages/496" /></p>

<p>Sketch a graph of </p><p>cosine function with an amplitude of 2, reflected in the x-axis.</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>

<p>A wheelchair ramp to the front porch
of a house is to be built so that it has an angle of inclination of 14.5° and a height
of 1.3 m.</p><p>Use a reciprocal trigonometric ratio
to write an equation that can be used
to determine the length of the ramp.</p>

<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions.
Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = 4\sin{x} +3</code></p>

<p> Sketch the following graph.</p><p><code class='latex inline'>
\displaystyle
y = 3\sin(\frac{1}{2}x )
</code></p>

<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions.
Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = -\sin\left(\displaystyle{\frac{1}{4}}x\right)</code></p>

<p>The graph shows the distance from a light pole to a car racing around a
circular track. The track is located north of the light pole.</p><img src="/qimages/2171" /><p>a) Determine the distance from the light pole to the edge of the track.</p>

<p>Determine an equation for each cosine function. </p><img src="/qimages/529" />

<p>Sketch a graph of </p><p>sine function with an amplitude of 4, reflected in the x-axis.</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>

<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p>
<ul>
<li>Graph the function over two cycles.</li>
</ul>

<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'>
\displaystyle
y = - \cos(2x + \frac{\pi}{8}) + 2
</code></p>

<p>A popular ride at many theme parks is the Gravitron, which pins riders to the walls of the ride as it spins and then drops the floor to leave them suspended by friction and centrifugal effects. Suppose that such a ride is designed with a radius of 4m and a period of 2s, with the origin at the centre of rotation. </p><p><strong>(a)</strong> Model the <code class='latex inline'>x</code>-position as a function of time <code class='latex inline'>t</code>, using a sine function.</p><p><strong>(b)</strong> Model the <code class='latex inline'>y</code>-position as a function of time, <code class='latex inline'>t</code>, using a cosine function.</p><p><strong>(c)</strong> Is it true that <code class='latex inline'>x^2+y^2=16</code> for all values of <code class='latex inline'>t</code>.</p>

<p>A sine function has a maximum value of 7, a minimum value of -1, a phase shift of <code class='latex inline'>\displaystyle{\frac{3\pi}{4}}</code> rad to the left, and a period of <code class='latex inline'>\displaystyle{\frac{\pi}{2}}</code>.</p><p>Write an equation for the function.</p>

<p> Sketch the following graph.</p><p><code class='latex inline'>
\displaystyle
y = -3\sin( \frac{2}{3}x )
</code></p>

<p>A pendulum swings back and forth 10 times in 8 s. It swings through
a total horizontal distance of 40 cm. </p><p>a) Sketch a graph of this motion for two cycles, beginning with the
pendulum at the end of its swing.</p>

<p>Recall the quotient identity <code class='latex inline'>\tan x = \frac{\sin x}{\cos x}</code>. Use the graphs of <code class='latex inline'>y = \sin x</code> and <code class='latex inline'>y= \cos x</code> in conjunction with the quotient identity to explain the shape of the graph of <code class='latex inline'>y = \tan x</code>.</p>

<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p>
<ul>
<li>Graph the model over two cycles.</li>
</ul>

<p>What is the maximum value? the minimum value? </p><img src="/qimages/980" />

<p>A crank 20 cm long, starting at <code class='latex inline'>60^{\circ}</code> above from the horizontal, sweeps out a positive angle at the uniform rate of 10 rad/s.</p><p>a) Determine the function that gives the height of the end of the crank from the horizontal line at any time t.</p>

<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following
sets of transformations. For each set of transformations, determine the
equation of the resulting function and sketch its graph.</p><p>vertical compression by a factor of <code class='latex inline'>\displaystyle{\frac{1}{2}}</code>, vertical translation 3 units up</p>

<p>Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At <code class='latex inline'>t = 0</code>, the sparkler is at its highest point above the ground.</p>
<ul>
<li>What does the equation of the axis of the sinusoidal function
represent in this situation?</li>
</ul>

<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>a) Determine the vertical translation of the function.</p><p>b) What is the desired period of the function?</p>

<p>A cosine function has a maximum value of 1, a minimum value of —5, a phase shift of 2 rad to the right, and a period of 3. </p><p> Write an equation for the function.</p>

<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions.
Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = \sin\left(4x+\displaystyle{\frac{2\pi}{3}}\right)</code></p>

<p>A cosine function has a maximum value of <code class='latex inline'>7</code> and minimum value of <code class='latex inline'>-3</code>.</p>
<ul>
<li>Determine the vertical translation.</li>
</ul>

<p>Write an equation for</p><p>cosine function with a phase shift of <code class='latex inline'>\frac{5\pi}{6}</code></p>

<p> Answer the following for each graph.</p>
<ul>
<li>amplitude</li>
<li>period</li>
</ul>
<p><em>(a)</em> <code class='latex inline'>
\displaystyle
y = 5\sin(x )
</code></p><p><em>(b)</em> <code class='latex inline'>
\displaystyle
y = -3 \sin(\frac{7}{2}x )
</code></p>

<p>Consider the function <code class='latex inline'>\displaystyle{=4\sin\left[2\left(x+\frac{2\pi}{3}\right)\right]-5}</code>. </p><p><strong>(a)</strong> What is the amplitude?</p><p><strong>(b)</strong> What is the period?</p><p><strong>(c)</strong> Describe the phase shift.</p><p><strong>(d)</strong> Describe the vertical translation.</p><p><strong>(e)</strong> Describe the <code class='latex inline'>x</code>-intercepts of the function. Explain your answer.</p>

<p>A nail is stuck in the tire of a car. If a student wanted to graph a sine function to model the height of the nail above the ground during a trip from London, Ontario, to Waterloo, Ontario, should the student graph the distance of the nail above the ground as a function of time or as a function of the total distance travelled by the nail? Explain your reasoning.</p>

<p>The graph shows the distance from a light pole to a car racing around a
circular track. The track is located north of the light pole.</p><img src="/qimages/2171" /><p>e) Determine the speed of the car in metres per second.</p>

<p> Sketch <code class='latex inline'>y = -\sin x</code>. Then sketch <code class='latex inline'>y = \sin(-x)</code>. What do you notice?</p>

<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions.
Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = \sin(x-\pi) - 1</code></p>

<p> Answer the following for each graph.</p>
<ul>
<li>amplitude</li>
<li>period</li>
</ul>
<p><code class='latex inline'>
\displaystyle
y = -2 \sin(3x )
</code></p>

<p>A contestant on a game show spins a wheel that is located on a plane perpendicular to the floor. He grabs the only red peg on the circumference of the wheel, which is 1.5 m above the floor, and pushes it downward. The red peg reaches a minimum height of 0.25 m above the floor and a maximum height of 2.75 m above the floor. Sketch two cycles of the graph that represents the height of the red peg above the floor, as a function of the total distance it moved. Then determine the equation of the sine function that describes the graph.</p>

<p>Write an equation for </p><p>sine function with an amplitude of 4, reflected in the x-axis.</p>

<p>What is the vertical translation? </p><img src="/qimages/980" />

<p>A popular attraction in many water them parks is the wave pool. Suppose that the wave generator moves back and forth 20 complete cycles every minute to produce waves measuring 1.2 m from the top of the crest to the bottom of the trough.</p>
<ul>
<li>Model the displacement, <code class='latex inline'>d</code> , in metres, of the wave from the mean pool level as a function of time, <code class='latex inline'>t</code>, in seconds.</li>
</ul>

<p>Model the graph shown using a sine function</p><img src="/qimages/527" /><p><strong>(a)</strong> From the graph, determine the amplitude, the period, the phase shift, and the vertical translation,</p><p><strong>(b)</strong> Write an equation for the function.</p>

<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'>
\displaystyle
y = \sin(2x + \frac{\pi}{8})
</code></p>

<p>A rung on a hamster wheel, with a radius of 25 cm, is travelling at a
constant speed. It makes one complete revolution in 3 s. The axle of
the hamster wheel is 27 cm above the ground.</p><p>b) Describe the transformations necessary to transform <code class='latex inline'>y = \cos{x}</code> into the function you graphed in part a).</p><p>c) Write the equation that models this situation.</p>

<p>Model the graph shown using a cosine function.</p><img src="/qimages/528" /><p><strong>(a)</strong> From the graph, determine the amplitude, the period, the phase shift, and the vertical translation,</p><p><strong>(b)</strong> Write an equation for the function.</p>

<p>State the period, amplitude, horizontal translation, and equation of
the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y=\sin\left(x-\displaystyle{\frac{\pi}{4}}\right)+3</code></p>

<p>The needle of a compass makes an angle of 4 radians with the line pointing east from the centre of the compass. The tip of the needle is 4.2 cm below the line pointing west from the centre of the compass. How long is the needle, to the nearest hundredth of a centimetre?</p>

<p>Write an equation for</p><p>cosine function with a phase shift of <code class='latex inline'>\frac{4\pi}{3}</code></p>

<p>The graph of a sinusoidal function has been vertically stretched, vertically
translated up, and horizontally translated to the right. The graph has a
maximum at <code class='latex inline'>\left(\displaystyle{\frac{\pi}{13}},13\right)</code>, and the equation of the axis is <code class='latex inline'>y = 9</code>. If
the x-axis is in radians, which point is the minimum of the graph?</p>

<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'>
\displaystyle
y = 2 \cos 3x + 10
</code></p>

<p>A crank 20 cm long, starting at <code class='latex inline'>60^{\circ}</code> above from the horizontal, sweeps out a positive angle at the uniform rate of 10 rad/s.</p>
<ul>
<li>At what times is the crank 12 cm from the horizontal line?</li>
</ul>

<p>A person who was listening to a siren reported that the frequency of the sound fluctuated with time, measured in seconds. The minimum frequency that the person heard was 500 Hz, and the maximum frequency was 1000 Hz. The maximum frequency occurred at <code class='latex inline'>t = 0</code> and <code class='latex inline'>t = 15</code>. The person also reported that, in 15, she heard the maximum frequency 6 times (including the times at <code class='latex inline'>t = 0</code> and <code class='latex inline'>t = 15</code>). What is the equation of the cosine function that describes the frequency of this siren?</p>

<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle
y = \sin x + 4
</code></p>

<p>State the period, amplitude, horizontal translation, and equation of
the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y = 5\cos\left(-2x+\displaystyle{\frac{\pi}{3}}\right)-2</code></p>

<p>The height of a patch on a bicycle tire above the ground, as a function of time, is modelled by one sinusoidal function. The height of the patch above the ground, as a function of the total distance it has travelled, is modelled by another sinusoidal function. Which of the following characteristics do the two sinusoidal functions share: amplitude, period, equation of the axis?</p>

<p>Write an equation for</p><p>sine function with an amplitude of 3</p>

<p>State the period, amplitude, horizontal translation, and equation of
the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y = 2\sin(3x)-1</code></p>

<p>The voltage, <code class='latex inline'>V</code>, in volts, applied to an electric circuit can be modelled by the equation <code class='latex inline'>V = 167 \sin(120\pi t)</code>, where <code class='latex inline'>t</code> is the time, in seconds. A component in the circuit can be safely withstand a voltage of more than<code class='latex inline'>120</code> V for <code class='latex inline'>0.01</code> s or less.</p><p> Is it safe to sue this component in this circuit? Just icy your answer.</p>

<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>sine function with a period of <code class='latex inline'>\frac{\pi}{2}</code></li>
</ul>

<p>A function is said to be even if <code class='latex inline'>f(-x) = f(x)</code> for all values of . A function is said to be odd if <code class='latex inline'>f(-x) = -f(x)</code> for all values of <code class='latex inline'>x</code>.</p>
<ul>
<li>Is the function <code class='latex inline'>y = \tan x</code> even, odd, or neither?</li>
</ul>

<p>A clock is showing the time as exactly 3:00 pm and 25 sec. Because a full minute has not passed since 3:00 pm, the hour hand is pointing directly at the 3 and the minute hand is pointing directly at the 12. If the tip of the second hand is directly below the tip of the hour hand, and if the length of the second hand is 9 cm, what is the length of the hour hand?</p>

<p>Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At <code class='latex inline'>t = 0</code>, the sparkler is at its highest point above the ground.</p>
<ul>
<li>What does the amplitude of the sinusoidal function represent in this situation?</li>
</ul>

<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing
technology.</p><p><code class='latex inline'>y = -\cos\left(0.5x-\displaystyle{\frac{\pi}{6}}\right)+3</code></p>

<p>The graph of a sinusoidal function has been horizontally compressed
and horizontally translated to the left. It has maximums at the points <code class='latex inline'>\left(-\displaystyle{\frac{5\pi}{7}},1\right)</code> and <code class='latex inline'>\left(-\displaystyle{\frac{3\pi}{7}},1\right)</code>, and it has a minimum at <code class='latex inline'>\left(-\displaystyle{\frac{4\pi}{7}},-1\right)</code>. If the x-axis is in radians, what is the period of the function?</p>

<p>A leaning flagpole, <code class='latex inline'>5 m</code> long, makes an obtuse angle with the ground. If the distance from the tip of the flagpole to the ground is <code class='latex inline'>3.4 m</code>, determine the radian measure of the obtuse angle, to the nearest hundredth.</p>

<p>For each of the following, state the amplitude, period, phase shift, and vertical translation. For each graph, state it as a sine function. Note that the period is set as a multiple of <code class='latex inline'>\pi</code></p><img src="/qimages/282" />

<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>(d) Determine the value of <code class='latex inline'>k</code> that results in the period desired in part (c).</p>

<p>Find the equation of the following function in terms of </p><p>a) a sine function and </p><p>b) a cosine function</p><img src="/qimages/2107" />

<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>sine function with a phase shift of <code class='latex inline'>-6\pi</code></li>
</ul>

<p>Sketch a graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<ul>
<li>cosine function with a phase shift of <code class='latex inline'>\frac{4\pi}{3}</code></li>
</ul>

<p>A cosine function has a maximum value of <code class='latex inline'>7</code> and minimum value of <code class='latex inline'>-3</code>.</p>
<ul>
<li>Graph the function over two cycles.</li>
</ul>

<p> Sketch the following graphs.</p><p><code class='latex inline'>
\displaystyle
\begin{array}{cccc}
&(a) & y = \sin(3x ) &(b) &y = \sin(5x ) \\
&(c) & y = \sin( \frac{1}{4}x ) &(d)&y = \sin( \frac{3}{5}x ) \\
\end{array}
</code></p>

<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following
sets of transformations. For each set of transformations, determine the
equation of the resulting function and sketch its graph.</p><p>horizontal stretch by a factor of 2, reflection in the y-axis</p>

<p>A sine function has an amplitude of 3 and a period of <code class='latex inline'>\pi</code>.</p><p><strong>(a)</strong> Write the equation of the function in the form <code class='latex inline'>y = a\sin kx</code>.</p><p><strong>(b)</strong> Graph the function over two cycles.</p>

<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p>
<ul>
<li>Determine the phase shift of the function.</li>
</ul>

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