Now You Try

<p>The observation deck at Justin's Cove lighthouse, in Prince Edward Island is about <code class='latex inline'>20</code> m above sea level. From the observation deck. the angle of depression of a boat on the water is <code class='latex inline'>6</code>°. How far is the boat from the lighthouse, to the nearest metre?</p>

<p>The Capilano Suspension Bridge in North Vancouver is the world’s highest footbridge of its kind. The bridge is 140 m long. From the ends of the bridge, the angles of depression of a point on the river under the bridge are 41° and 48°. How high is the bridge above the river, to the nearest metre?</p>

<p> Find the tangent of the other acute angle, to four decimal places. </p><img src="/qimages/2374" />

<p>Find each trigonometric ratio for angle <code class='latex inline'>A</code> in the triangle at the right. </p><img src="/qimages/60797" /><p><code class='latex inline'>\displaystyle \tan A </code></p>

<p>Determine the value of <code class='latex inline'>\theta</code>, to the nearest degree, in each triangle.</p><p> <img src="/qimages/1033" /></p>

<p>Decide whether each statement is true
or false. Justify your decision.</p><p><code class='latex inline'>\sin \theta = 0.4</code></p><img src="/qimages/1023" />

<p> Find the tangent of the other acute angle, to four decimal places. </p><img src="/qimages/2375" />

<p>The resulting value of <code class='latex inline'>\sin, \cos, \tan</code> of an angle could be? (circle all that could be true)</p><p><code class='latex inline'>
\begin{array}{cccccc}
&(a) &\text{Angle } &(b)& \text{ decimal } &(c)& \text{ ratio }\\
&(d) & \text{ speed }
&(e)& \text{ distance } \\
\end{array}
</code></p>

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre. </p><img src="/qimages/2387" />

<p>Find the length of the unknown side, to the nearest tenth.</p><img src="/qimages/2380" />

<p>Find the measures of both acute angles in each triangle, to the nearest degree. </p><img src="/qimages/2378" />

<p>Draw two different right triangles for which <code class='latex inline'>\tan\theta=1</code>. Determine the measurements of all the sides and angles. Then compare
the two triangles.</p>

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre. </p><img src="/qimages/2382" />

<p>Find the tangent of the angle indicated to four decimal places.</p><img src="/qimages/2373" />

<p> Find the tangent of the other acute angle, to four decimal places. </p><img src="/qimages/2373" />

<p>In the diagram, <code class='latex inline'>\triangle ABC</code> is isosceles, with <code class='latex inline'>AB = AC</code>, and <code class='latex inline'>\triangle RST</code> is equilateral. Express <code class='latex inline'>\angle x</code> in terms of <code class='latex inline'>\angle y</code> and <code class='latex inline'>\angle z</code></p><img src="/qimages/5597" />

<p>The shadow of a tree that is 12 m tall measures 9 m in length. Determine the angle of elevation of the sun.</p>

<p> Find the tangent of the other acute angle, to four decimal places. </p><img src="/qimages/2372" />

<p>A scuba diver swam north at 1.5 m/s. across a current running from east to west at 2.0 m/s. She swam for 3 min and then surfaced.</p><p>a) Draw a diagram showing where the dive boat will pick her up relative to where she dove.</p><p>b) How far did she travel?</p>

<p>In a right</p><p>triangle, the hypotenuse</p><p>is 5 in. long, and the side</p><p>opposite <code class='latex inline'>\displaystyle \angle A </code> is <code class='latex inline'>\displaystyle 4.5 </code> in. long. A student found the</p><p>measure of <code class='latex inline'>\displaystyle \angle A </code> as <code class='latex inline'>
\displaystyle
sin(0.9) = 0.0157
</code></p><p> Describe and correct the student's error.</p>

<p> Triangle ABC has a base BC = 115 cm, side BA = 80 cm, and <code class='latex inline'>\angle B = 35^{\circ}</code>. Determine the height h from the vertex A and the area of <code class='latex inline'>\triangle ABC.</code> Leave your answer in exact value.</p>

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre. </p><img src="/qimages/2383" />

<p>Find the tangent of the angle indicated to four decimal places.</p><img src="/qimages/2370" />

<p>A bridge is going to be built across a river. To determine the width of
the river, a surveyor on one bank sights the top of a pole, which is 3 m
high, on the opposite bank. His optical device is mounted 1.2 m above
the ground. The angle of elevation to the top of the pole is 85<code class='latex inline'>^{\circ}</code>. How
wide is the river?</p>

<p>Find the tangent of the angle indicated to four decimal places.</p><img src="/qimages/2372" />

<p>Find the missing values: <strong>a</strong> and <strong>b</strong>. Leave your answer in exact value.</p><img src="/qimages/248" />

<p>For <code class='latex inline'>\displaystyle \triangle F G H </code> and <code class='latex inline'>\displaystyle \triangle L M N </code>, find the value of each expression.</p><img src="/qimages/60806" /><p><code class='latex inline'>\displaystyle \sin F </code></p>

<p>For <code class='latex inline'>\displaystyle \triangle F G H </code> and <code class='latex inline'>\displaystyle \triangle L M N </code>, find the value of each expression.</p><img src="/qimages/60806" /><p><code class='latex inline'>\displaystyle \tan G </code></p>

<p>Find each trigonometric ratio for angle <code class='latex inline'>A</code> in the triangle at the right. </p><img src="/qimages/60797" /><p><code class='latex inline'>\displaystyle \sin A </code></p>

<p>Decide whether each statement is true or false. Justify your decision.</p><p><code class='latex inline'>\tan \alpha = 2</code></p><img src="/qimages/1023" />

<p>Giulia unreels her kite string until the kite is flying on a string of length 50m. </p><p>A light breeze holds the kite such that the sting make and angle of <code class='latex inline'>60^o</code> with the ground. After a few minutes, the wind pickup speed. The wind now pushes the kite until the string makes an angle of <code class='latex inline'>45^o</code> with the ground. </p><p>What distance has the kite changed position horizontally over the ground. Determine an exact expression. </p>

<p>Does <code class='latex inline'>\cos60^{\circ}=\displaystyle{\frac{1}{2}}</code> mean that the side adjacent to the <code class='latex inline'>60^{\circ}</code> angle measure 1 unit and the hypotenuse measure 2 units? Explain.</p>

<p>To photograph a rocket stage separating, Lucien mounts his camera
on a tripod. The tripod can be set to the angle at which the stage will
separate. This is where Lucien needs to aim his lens. He begins by
aiming his camera at the launch pad, which is 1500 m away. The stage
will separate at 20 000 m. At what angle should Lucien set the tripod?</p>

<p> Given that <code class='latex inline'>\angle B = 90^{\circ}</code> and <code class='latex inline'>\cot C = \frac{5}{6}</code> in <code class='latex inline'>\triangle ABC</code>, find the side BC if <code class='latex inline'>AC = 5\sqrt{61}</code>. Leave your answer in exact value.</p>

<p>Solve for <code class='latex inline'>x</code>, and express your answer to one decimal place.</p><p> <code class='latex inline'>\tan75^{\circ}=\displaystyle{\frac{x}{20}}</code></p>

<p>A telephone pole is secured at its top with a guy wire, as shown. The guy wire makes an angle of <code class='latex inline'>70</code>° with the ground and is secured <code class='latex inline'>5.6</code> m from the bottom of the pole. Find the height of the telephone pole.</p>
<ul>
<li>Find the length of the guy wire using two different methods.</li>
</ul>
<img src="/qimages/1618" />

<p>Solve for <code class='latex inline'>\angle A</code> to the nearest degree.</p><p><code class='latex inline'>\sin A=0.9063</code></p>

<p>Determine the diameter of the circle, if <code class='latex inline'>O</code> is the centre of the circle.</p><img src="/qimages/1046" />

<p>Find the missing values: <strong>a</strong> and <strong>b</strong>. Leave your answer in exact value.</p><img src="/qimages/249" />

<p>Find the tangent of the angle indicated to four decimal places.</p><img src="/qimages/2375" />

<p>The Great Pyramid of Cheops is a square-based pyramid with a height of 147 m and a base length of 230 m. Find the angle, <code class='latex inline'>\theta</code>, to the nearest degree, that one of the edges of the pyramid makes with the base.</p><img src="/qimages/5492" />

<p>Find the measures of both acute angles in each triangle, to the nearest degree. </p><img src="/qimages/2377" />

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre.</p><img src="/qimages/2389" />

<p>Each side length of regular pentagon <code class='latex inline'>ABCDE</code> is 8.2 cm.</p><p>Calculate the measure of <code class='latex inline'>\theta</code> to the nearest degree.</p><img src="/qimages/1054" />

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre.</p><img src="/qimages/2388" />

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre. </p><img src="/qimages/2386" />

<p>Determine the angle between the line <code class='latex inline'>y=\displaystyle{\frac{3}{2}}x+4</code> and the x-axis.</p>

<p>Find the length of the unknown side, to the nearest tenth.</p><img src="/qimages/2382" />

<p>For a ladder to be stable, the angle that it makes with the ground
should be no more than 78<code class='latex inline'>^{\circ}</code> and no less than 73<code class='latex inline'>^{\circ}</code>.</p>
<ul>
<li>If the base of a ladder that is 8.0 m long is placed 1.5 m from a wall, what is the angle between the ground and the ladder? Is this safe?</li>
</ul>

<p>Find the measures of both acute angles in each triangle, to the nearest degree. </p><img src="/qimages/2376" />

<p>If in <code class='latex inline'>\triangle ABC</code>, <code class='latex inline'>AC = b, BC = a</code>, and <code class='latex inline'>\angle C</code> is known, then the area of <code class='latex inline'>\triangle ABC = \frac{1}{2}ab \sin C</code>. Show that this formula is true.</p>

<p>Decide whether each statement is true
or false. Justify your decision.</p><p> <code class='latex inline'>\cos\alpha \doteq 0.8929</code></p><img src="/qimages/1023" />

<p>Find the tangent of the angle indicated to four decimal places.</p><img src="/qimages/2371" />

<p> A perpendicular <code class='latex inline'>\overline{AP}</code> is drawn to base <code class='latex inline'>\overline{BC}</code> in <code class='latex inline'>\triangle ABC</code>. If <code class='latex inline'>\overline{AP} = 12, \overline{BP} = 5</code>, and <code class='latex inline'>\overline{PC} = 25</code>, determine if <code class='latex inline'>\triangle ABC</code> contains a right angle. Leave your answer in exact value.</p><img src="/qimages/2358" />

<p>For a ladder to be stable, the angle that it makes with the ground
should be no more than 78<code class='latex inline'>^{\circ}</code> and no less than 73<code class='latex inline'>^{\circ}</code>.</p><p> What are the minimum and maximum safe distances from
the base of the ladder to the wall?</p>

<p>An octahedron is formed by attaching eight congruent equilateral triangles, as shown.</p><img src="/qimages/5489" /><p>If the length along one of the edges is
20 cm, find the distance between opposite vertices.</p>

<p>Dennis has let out <code class='latex inline'>40 m</code> of his kite string, which makes an angle of <code class='latex inline'>72^o</code> with the horizontal ground.</p><img src="/qimages/4024" /><p>a) Find the height of the kite, to the nearest metre.</p><p>b) Suppose the Sun is shining directly above the kite. How far is the kite's shadow from Dennis, to the nearest meter?</p>

<p>Solve for <code class='latex inline'>x</code>, and express your answer to one decimal place.</p><p><code class='latex inline'>\tan80^{\circ}=\displaystyle{\frac{12}{x}}</code></p>

<p>Calculate the measure of <code class='latex inline'>x</code> in the diagram to the nearest
degree, using one of the primary trigonometric ratios.</p><img src="/qimages/1024" />

<p>Do you need to use a primary trigonometric ratio to determine
the measure of <code class='latex inline'>y</code>? Explain.</p><img src="/qimages/1024" />

<p>Identify the primary trigonometric ratio for <code class='latex inline'>\theta</code> that is equal to each ratio for the triangle.</p><p><strong>a)</strong> <code class='latex inline'>\displaystyle{\frac{50}{54}}</code></p><p><strong>b)</strong> <code class='latex inline'>\displaystyle{\frac{20}{50}}</code></p><p><strong>c)</strong> <code class='latex inline'>\displaystyle{\frac{20}{54}}</code></p><img src="/qimages/1025" />

<p>To measure the width of a river, Kristie uses a large rock, and oak tree, and an elm tree, which are positioned as shown.</p><img src="/qimages/2390" /><p>Show how Kristie can use the tangent ratio to find the width of the river, to the nearest metre. </p>

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre. </p><img src="/qimages/2385" />

<p>Find each trigonometric ratio for angle <code class='latex inline'>A</code> in the triangle at the right. </p><img src="/qimages/60797" /><p><code class='latex inline'>\displaystyle \cos A </code></p>

<p>Three blood stains from a victim are of the aircraft, so that the beams will meet shown. The point of convergence, C, has been found by extrapolating the directions
of these stains along the floor. The origin of the blow, O, is some height above C.</p><img src="/qimages/5596" /><p>Forensic analysis of Stain 1 provides the following data.</p>
<ul>
<li>Length of bloodstain: 4.2 cm</li>
<li>Width of bloodstain: 2.6 cm</li>
<li>Distance from point of convergence: 2.1</li>
</ul>
<p>Determine the height at which the blow struck the victim.</p>

<p>For <code class='latex inline'>\displaystyle \triangle F G H </code> and <code class='latex inline'>\displaystyle \triangle L M N </code>, find the value of each expression.</p><img src="/qimages/60806" /><p><code class='latex inline'>\displaystyle \cos L </code></p>

<p>Determine each ratio, and write it as a decimal to four decimal places.</p><p><strong>a)</strong> <code class='latex inline'>\sin C</code></p><p><strong>b)</strong> <code class='latex inline'>\cos C</code></p><p><strong>c)</strong> <code class='latex inline'>\tan B</code></p><p><strong>d)</strong> <code class='latex inline'>\tan C</code></p><p><strong>e)</strong> <code class='latex inline'>\cos B</code></p><p><strong>f)</strong> <code class='latex inline'>\sin B</code></p><img src="/qimages/2409" />

<p> A group of students measured a distance of 50 cm from the base of a pole CB to a point A. If <code class='latex inline'>\angle CAB = 16.5^{\circ}</code>, calculate the height of the pole, correct to one decimal. Leave your answer in exact value.</p><img src="/qimages/247" />

<p>Evaluate with a calculator. Record your answer to four decimal places. </p><p><code class='latex inline'>
\displaystyle
\begin{array}{ccccccc}
& a)& \tan65^\circ & b) &\tan15^\circ & c) & \tan62^\circ & d) & \tan30.7^\circ \\
& e)& \tan82.4^\circ & f)& \tan82.4^\circ & g) & \tan20.5^\circ & h) & \tan45^\circ \\
\end{array}
</code></p>

<p>For what value of <code class='latex inline'>\theta</code> does <code class='latex inline'>\sin\theta=\cos\theta</code>? Include a diagram in your answer.</p>

<p>After <code class='latex inline'>1 h</code>, an airplane has travelled 350 km. Strong winds, however,
have caused the plane to be 48 km west of its planned flight path.
By how many degrees is the airplane off its planned flight path?</p><img src="/qimages/1050" />

<p>Determine the acute angle at which <code class='latex inline'>y=2x-1</code> and <code class='latex inline'>y=0.5x+2</code> intersect.</p>

<p>Rocco and Biff are two Koalas sitting at the top of two eucalyptus trees, which are located 10 m apart, as shown. Rocco's tree is exactly half as tall as Biff's tree. From Rocco's point of view, the angle separating Biff and the base of his tree is 70<code class='latex inline'>^\circ</code></p><img src="/qimages/2392" /><p>How high off the ground is each koala?</p>

<p>An isosceles triangle has a height of 12.5 m (measured from the
unequal side) and two equal angles that measure 55<code class='latex inline'>^{\circ}</code>. Determine
the area of the triangle.</p>

<p>Determine the value of <code class='latex inline'>\theta</code>, to the nearest degree, in each triangle.</p><img src="/qimages/1032" />

<p>Solve using trigonometric ratios. </p><p>A right triangle's legs are 7 in. and 24 in. long. What is the measure of the angle opposite the 24 -in. leg?</p>

<p>Using trigonometry, calculate the measures of <code class='latex inline'>\angle A</code> and <code class='latex inline'>\angle B</code> in each triangle. Round your answers to the nearest degree.</p><p> <img src="/qimages/1038" /></p>

<p>Elise drew a diagram of her triangular yard. She wants to cover her yard with sod. Explain how you could calculate the cost, if sod
costs $1.50/m<code class='latex inline'>^2</code>.</p><img src="/qimages/1052" />

<p>Explain why the value of <code class='latex inline'>\tan\theta</code> increases as the measure of <code class='latex inline'>\theta</code> increases.</p><img src="/qimages/1029" />

<p> Find the tangent of the other acute angle, to four decimal places. </p><img src="/qimages/2370" />

<p> Find the tangent of the other acute angle, to four decimal places. </p><img src="/qimages/2371" />

<p>Find the length of the unknown side, to the nearest tenth.</p><img src="/qimages/2381" />

<p>Describe the difference between finding the sine of an angle and the cosine of an angle.</p>

<p>Find the measures of both acute angles in each triangle, to the nearest degree. </p><img src="/qimages/2379" />

<p>Use geometric reasoning to show that the angle of impact can be found using the relationship <code class='latex inline'>\sin \theta = \frac{d}{BC}</code>.</p>

<p>If the value of <code class='latex inline'>\sin \theta = \frac{2}{5}</code>, find the length of adjacent to <code class='latex inline'>\theta</code> if the opposite side to <code class='latex inline'>\theta</code> is 8. Leave your answer in exact value.</p>

<p>Determine the length of <code class='latex inline'>x</code>. Then state the primary trigonometric ratios for <code class='latex inline'>\theta</code>.</p><p> <img src="/qimages/1026" /></p>

<p>A surveyor is positioned at a traffic intersection, viewing a marker on the other side of the street. The marker is 19 m from the intersection. The surveyor cannot measure the width directly because there is too much traffic. Find the width of James Street, to the nearest tenth of a metre. </p><img src="/qimages/2391" />

<p>Find the value of <code class='latex inline'>x</code> to the nearest tenth. </p><img src="/qimages/60828" />

<p>Solve using trigonometric ratios. </p><p>A right triangle has a <code class='latex inline'>\displaystyle 40^{\circ} </code> angle. The hypotenuse is <code class='latex inline'>\displaystyle 10 \mathrm{~cm} </code> long. What is the length of the side opposite the <code class='latex inline'>\displaystyle 40^{\circ} </code> angle?</p>

<p>Find the measure of each angle, to the nearest degree. </p><p><code class='latex inline'>
\displaystyle
\begin{array}{ccccccc}
& a)& \tan\theta=1.5 & b) & \tan A=\frac{3}{4} & c) & \tan B=0.6000 & d) & \tan W=\frac{4}{5}\\
& e)& \tan C=0.8333 & f)& \tan\theta=\frac{6}{7} & g) & \tan X=3.0250 & h) & \tan\theta=\frac{15}{9}
\end{array}
</code></p>

<p> Find the value of <code class='latex inline'>\sin \theta</code> and <code class='latex inline'>\cos \theta</code> if <code class='latex inline'>\tan \theta = \frac{5}{7}</code>. Leave your answer in exact value.</p>

<p>Find the tangent of the angle indicated to four decimal places.</p><img src="/qimages/2374" />

<p>Use the definitions of sine, cosine, and tangent to simplify each expression.</p><p><code class='latex inline'>\displaystyle \begin{array}{lll}\text { a. } \cos A \cdot \tan A & \text { b. } \sin A \div \tan A & \text { c. } \sin A \div \cos A\end{array} </code></p>

<p>Decide whether each statement is true
or false. Justify your decision.</p><p><code class='latex inline'>\cos \theta \doteq 0.8929</code></p><img src="/qimages/1023" />

<p>A guy wire is attached to a cellphone tower as shown at the left.
The guy wire is 30 m long, and the cellphone tower is 24 m high.
Determine the angle that is formed by the guy wire and the ground.</p><img src="/qimages/1049" />

<p>Solve for <code class='latex inline'>x</code>, and express your answer to one decimal place.</p><p> <code class='latex inline'>\cos45^{\circ}=\displaystyle{\frac{x}{6}}</code></p>

<p>To get to school, Eli can travel 1.2 km east on Rutherford St. and then south on Orchard Av. to his school. Or, he can take a shortcut through the park, as shown. His shortcut takes him 20 min.</p><p>Eli's walking speed is 6 km/h.</p><p>a) What angle does Eli shortcut make with Rutherford st.? Describe any assumptions you must make.</p><p>b) How much time does Eli save by taking his shortcut?</p><img src="/qimages/4025" />

<p> In a circle O with radius 6, <code class='latex inline'>\angle AOB = 60^{\circ}</code> and <code class='latex inline'>\angle COD = 90^{\circ}</code>. Find the difference in the lengths of line segments CD and AB. Leave your answer in exact value.</p>

<p>Find the length of <code class='latex inline'>x</code>, to the nearest tenth of a metre. </p><img src="/qimages/2384" />

<p><strong>a)</strong> Which side is opposite to <code class='latex inline'>\angle A</code>?</p><p><strong>b)</strong> Which side is adjacent to <code class='latex inline'>\angle A</code>?</p><p><strong>c)</strong> Which side is the hypotenuse? </p><img src="/qimages/2408" />

<p>If <code class='latex inline'>\tan \theta = \frac{4}{5}</code>, which one of of the following statements must be true?</p><p><strong>(a)</strong> Hypotenuse is double the length of Adjacent </p><p><strong>(b)</strong> Opposite is longer than Adjacent </p><p><strong>(c)</strong> Opposite is shorter than Adjacent </p><p><strong>(d)</strong> Opposite is triple the length of Adjacent </p><p><strong>(e)</strong> Hypotenuse is longer than the sum of Adjacent and Opposite </p>

<p>Determine the length of <code class='latex inline'>x</code>. Then state the primary trigonometric ratios for <code class='latex inline'>\theta</code>.</p><p> <img src="/qimages/1027" /></p>

<p>Each side length of regular pentagon <code class='latex inline'>ABCDE</code> is 8.2 cm.</p><p> Calculate the length of diagonal <code class='latex inline'>AC</code> to the nearest tenth
of a centimetre.</p><img src="/qimages/1054" />

<p>For <code class='latex inline'>\displaystyle \triangle F G H </code> and <code class='latex inline'>\displaystyle \triangle L M N </code>, find the value of each expression.</p><img src="/qimages/60806" /><p><code class='latex inline'>\displaystyle \cos F </code></p>

<p>Solve for <code class='latex inline'>x</code>, and express your answer to one decimal place.</p><p> <code class='latex inline'>\sin62^{\circ}=\displaystyle{\frac{x}{14}}</code></p>