Now You Try

<p>Convert to degree measure.</p><p><code class='latex inline'>-\displaystyle{\frac{9\pi}{2}}</code></p>

<p>Two pieces of mud are stuck to the spoke of a bicycle wheel. Piece A is closer to the circumference of the tire, while piece B is closer to the centre of the wheel.</p><p>Is the angular velocity at which piece A is traveling greater than, less than, or equal to the angular velocity at which piece B is traveling?</p>

<p>Determine the approximate radian measure, to the nearest hundredth, for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
128^{\circ}
</code></li>
</ul>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
40^{\circ}
</code></li>
</ul>

<p> The measures of two angles in standard position are given. Determine whether the angles are co-terminal(in the same position in the circle).</p><p><code class='latex inline'>
\displaystyle
\frac{5\pi}{6}, \frac{17\pi}{6}
</code></p>

<p>Determine the approximate radian measure, to the nearest hundredth, for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
330^{\circ}
</code></li>
</ul>

<p>Convert to degree measure.</p><p> <code class='latex inline'>-\displaystyle{\frac{3\pi}{2}}</code></p>

<p> Find the degree measure of the angle with the given radian measure.</p><p>(a) <code class='latex inline'>\frac{7\pi}{6}</code></p><p>(b) <code class='latex inline'> -\frac{5 \pi}{6}</code></p>

<p>Determine the approximate degree measure, to the nearest tenth, for each angle.</p>
<ul>
<li>2.34 rad</li>
</ul>

<p>The Moon has a diameter of about 3480 km and an orbital radius of about 384 400 km from the centre of Earth. Suppose that the Moon is directly overhead. What is the measure of the angle subtended by the diameter to the Moon as measured by an astronomer on the surface of Earth? Answer in both radians and degrees. NOTE: the radius of the earth is 6371 km.</p>

<p> Find the radian measure of the angle with the given degree measure.</p><p><code class='latex inline'>
\displaystyle
-300^{\circ}
</code></p>

<p>Determine the exact degree measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
\frac{5\pi}{12}
</code></li>
</ul>

<p>The distance between two adjacent seats in a ferris wheel is about 6 m. If the angle measure from the centre of the ferris wheel is 0.2 rad, find the length of the radius of the ferris wheel.</p>

<p>The London Eye is a large Ferris wheel located on the banks of the Tames River in London, England. Each sealed and air-conditioned passenger capsule holds about 25 passengers. The diameter of the wheel is 135 m, and the wheel takes about half an hour to complete one revolution.</p><p><strong>(a)</strong> Determine the exact angle, in radians, that a passenger will travel in 5 min.</p><p><strong>(b)</strong> How far does a passenger travel in 5 min?</p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>45^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{4}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
90^{\circ}
</code></li>
</ul>

<p>Convert to degree measure.</p><p> <code class='latex inline'>-\displaystyle{\frac{3\pi}{4}}</code></p>

<p>Convert to radian measure.</p><p>240<code class='latex inline'>^{\circ}</code></p>

<p>Determine the approximate degree measure, to the nearest tenth, for each angle.</p>
<ul>
<li><code class='latex inline'>5.27</code></li>
</ul>

<p>Convert to degree measure.</p><p><code class='latex inline'>\displaystyle{\frac{7\pi}{6}}</code></p>

<p>Convert to radian measure.</p><p> <code class='latex inline'>-120^{\circ}</code></p>

<p> Find the radian measure of the angle with the given degree measure.</p><p><code class='latex inline'>
\displaystyle
3960^{\circ}
</code></p>

<p>Two pieces of mud are stuck to the spoke of a bicycle wheel. Piece A is closer to the circumference of the tire, while piece B is closer to the centre of the wheel.</p><p>If the angular velocity of the bicycle wheel increased, would the velocity at which piece A is traveling as a percent of the velocity at which piece B is traveling increase, decrease, or stay the same?</p>

<p>Two highways meet at an angle measuring <code class='latex inline'>\frac{\pi}{3}</code> rad, as shown. An on-ramp in the same of a circular arc in to be built such that the arc has a radius of 80.</p><img src="/qimages/8173" /><p><strong>(a)</strong> Determine an exact expression for the length of the on-ramp.</p><p><strong>(b)</strong> Determine the length of the on-ramp, to the nearest tenth of a metre.</p>

<p> Find the radian measure of the angle with the given degree measure.</p><p><code class='latex inline'>
\displaystyle
1080^{\circ}
</code></p>

<p>b) Determine the arc length of the circle in part a) if the central angle is <code class='latex inline'>200^{\circ}</code>.</p>

<p>Determine the arc length of a circle with a radius of 8 cm if
the central angle is 3.5 </p>

<p> Find the radian measure of the angle with the given degree measure.</p><p><code class='latex inline'>
\displaystyle
-75^{\circ}
</code></p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>45^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{4}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
135^{\circ}
</code></li>
</ul>

<p> Find the degree measure of the angle with the given radian measure.</p><p><code class='latex inline'>
\displaystyle
-\frac{13\pi}{12}
</code></p>

<p>Aircraft and ships use nautical miles for meaning distances. At one time, a nautical mile was defined as one minute of arc, or <code class='latex inline'>\frac{1}{60}</code> of a degree, along a meridian of longitude, following Earth's surface. </p>
<ul>
<li>The radius of Earth is about 6400 km. Determine the length of a nautical mile, using the old definition, to the nearest metre.</li>
</ul>
<p><a href="https://youtu.be/DP2U9SasYBk">HINT</a></p>

<p>Determine the exact degree measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
\frac{3\pi}{4}
</code></li>
</ul>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
9^{\circ}
</code></li>
</ul>

<p> The measures of two angles in standard position are given. Determine whether the angles are co-terminal(in the same position in the circle).</p><p><code class='latex inline'>
\displaystyle
-30^{\circ}, 330^{\circ}
</code></p>

<p>If a circle has a radius of 65 m, determine the arc length for each of the following central angles.</p><p> <code class='latex inline'>\displaystyle{\frac{19\pi}{20}}</code></p>

<p>Circle A has a radius of 15 cm and a central angle of <code class='latex inline'>\displaystyle \frac{\pi}{6}</code> radians, circle B has a radius of 17 cm and a central angle of <code class='latex inline'>\displaystyle \frac{\pi}{7}</code> radians, and circle C has a radius of 14 cm and a central angle of <code class='latex inline'>\displaystyle \frac{\pi}{5}</code> radians. </p><p>Put the circles in order, from smallest to largest, based on the lengths of the arcs subtending the central angles.</p>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
75^{\circ}
</code></li>
</ul>

<p>A length of radius of a bicycle wheel is 0.5 m. If the wheel turned to travel 12m, how much did the bicycle turn in radians?</p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\dfrac{\pi}{6}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
60^{\circ}
</code></li>
</ul>

<p>Determine the exact degree measure for each angle.</p>
<ul>
<li> <code class='latex inline'>
\displaystyle
\frac{\pi}{5}
</code></li>
</ul>

<p>Convert to degree measure.</p><p> <code class='latex inline'>\displaystyle{\frac{11\pi}{6}}</code></p>

<p>Convert to radian measure.</p><p>45<code class='latex inline'>^{\circ}</code></p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\dfrac{\pi}{6}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
90^{\circ}
</code></li>
</ul>

<p>Determine the approximate radian measure, to the nearest hundredth, for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
51^{\circ}
</code></li>
</ul>

<p>Kumar rides his bicycle such that the back wheel rotates 10 times in 5 s. Determine the angular velocity of the wheel in :</p><p>degrees per second</p><p>radians per second</p>

<p> The measures of two angles in standard position are given. Determine whether the angles are co-terminal(in the same position in the circle).</p><p><code class='latex inline'>
\displaystyle
70^{\circ}, 430^{\circ}
</code></p>

<p>A racing car driver travels in a circular course around a judges' stand. A distance of 1 km on the track subtends an angle of <code class='latex inline'>120^o</code> at the judges' stand. Determine the diameter of the track.</p>

<p> Find the radian measure of the angle with the given degree measure.</p><p><code class='latex inline'>
\displaystyle
202.5^{\circ}
</code></p>

<p>A milliradian (mrad) is <code class='latex inline'>\frac{1}{1000}</code> of a radian. Milliradians are used in artillery to estimate the distance to a target.</p>
<ul>
<li>Show that an arc of length 1 m subtends an angle of 1 mrad at a distance of 1 km.</li>
</ul>

<p>Convert to degree measure.</p><p> <code class='latex inline'>\displaystyle{\frac{\pi}{4}}</code></p>

<p>Sketch each rotation about a circle of radius 1.</p><p> <code class='latex inline'>\pi</code></p>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p> <img src="/qimages/494" /></p>

<p>Aircraft and ships use nautical miles for meaning distances. At one time, a nautical mile was defined as one minute of arc, or <code class='latex inline'>\frac{1}{60}</code> of a degree, along a meridian of longitude, following Earth's surface. </p><p>(a) Determine the radian measure of one minute of arc, to six decimal places.</p>

<p>Determine the exact degree measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
\frac{3\pi}{2}
</code></li>
</ul>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{6}</code> rad.</p><p><code class='latex inline'>
\displaystyle
10^{\circ}
</code></p>

<p>Two pieces of mud are stuck to the spoke of a bicycle wheel. Piece A is closer to the circumference of the tire, while piece B is closer to the centre of the wheel.</p><p>Is the velocity at which piece A is traveling greater than, less than, or equal to the velocity at which piece B is traveling?</p>

<p>A wheel is rotating at an angular velocity of <code class='latex inline'>1.2\pi</code> radians/s, while a point on the circumference of the wheel travels <code class='latex inline'>9.6\pi</code> m in <code class='latex inline'>10</code> s.</p><p>How many revolutions does the wheel make in 1 min?</p><p>What is the radius of the wheel?</p>

<p>If the area of a sector is <code class='latex inline'>36 \pi</code>, and the radius is 12, find the central angle of the sector in radians.</p>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
210^{\circ}
</code></li>
</ul>

<p>Circle A has a radius of 15 cm and a central angle of <code class='latex inline'>\frac{\pi}{6}</code> radians, circle B has a radius of 17 cm and a central angle of <code class='latex inline'>\frac{\pi}{7}</code> radians, and circle C has a radius of 14 cm and a central angle of <code class='latex inline'>\frac{\pi}{5}</code> radians. Put the circles 5 in order, from smallest to largest, based on the lengths of the arcs subtending the central angles.</p>

<p>A wind turbine has three blades, each measuring 3 m from centre to tip. At a particular time, the turbine is rotating four times a minute. </p><p>Determine the angular velocity of the turbine in radians/second.</p><p>How far has the tip of a blade traveled after 5 min?</p>

<p>Determine the approximate radian measure, to the nearest hundredth, for each angle.</p>
<ul>
<li> <code class='latex inline'>
\displaystyle
23^{\circ}
</code></li>
</ul>

<p> Find the degree measure of the angle with the given radian measure.</p><p><code class='latex inline'>
\displaystyle
3
</code></p>

<p>If a circle has a radius of 65 m, determine the arc length for each of the following central angles.</p><p>1.25</p>

<p>A fly wheel has a radius of <code class='latex inline'>3</code> m and is turning at a rate of <code class='latex inline'>300</code> rpm.</p><p>Determine the angular velocity of the wheel in radians per second.</p><p>Determine the linear velocity, in meters per second of the belt which drives the wheel.</p>

<p>An engine on a jet aircraft turns at about <code class='latex inline'>12 000</code> rpm. </p><p>Find an exact value, as well as an approximate value, for the angular velocity of the engine in radians per second.</p>

<p>Convert to radian measure.</p><p>270<code class='latex inline'>^{\circ}</code></p>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>g) <img src="/qimages/493" /></p>

<p>A wheel traveling at <code class='latex inline'>10</code> rpm covers <code class='latex inline'>33</code> linear feet in <code class='latex inline'>1</code> minute. Determine the radius of the wheel.</p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>45^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{4}</code> rad.</p><p><code class='latex inline'>
\displaystyle
225^{\circ}
</code></p>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
315^{\circ}
</code></li>
</ul>

<p>A milliradian (mrad) is <code class='latex inline'>\frac{1}{1000}</code> of a radian. Milliradians are used in artillery to estimate the distance to a target.</p>
<ul>
<li>A target scope shows that a target known to be 2m high subtends an angle of 0.25 mad. How far away is the target, in kilometres?</li>
</ul>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{6}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
7.5^{\circ}
</code></li>
</ul>

<p>Convert to degree measure.</p><p><code class='latex inline'>\displaystyle{\frac{2\pi}{3}}</code></p>

<p> Find the degree measure of the angle with the given radian measure.</p><p><code class='latex inline'>
\displaystyle
\frac{\pi}{10}
</code></p>

<p>Determine the exact degree measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
\frac{5\pi}{18}
</code></li>
</ul>

<p>Convert to radian measure.</p><p> 90<code class='latex inline'>^{\circ}</code></p>

<p>Determine the approximate radian measure, to the nearest hundredth, for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
240^{\circ}
</code></li>
</ul>

<p>Determine the length of <code class='latex inline'>AB</code>. Find the sine, cosine, and tangent ratios of <code class='latex inline'>\angle D</code>, given <code class='latex inline'>AC = CD = 8 cm</code>.</p><img src="/qimages/984" />

<p>Determine the area watered by a sprinkler system with a range of 200 m and a turning angel of <code class='latex inline'>\frac{3\pi}{4}</code></p>

<p> Find the area of the shaded region in the figure.</p><img src="/qimages/280" />

<p>The members of a high-school basketball team are driving from Calgary to Vancouver, which is a distance of 675 km. Each tire on their van has a radius of 32 cm. </p><p>If the team members drive at a constant speed and cover the distance from Calgary to Vancouver in 6 h 45 min, what is the angular velocity, in radians/second, of each tire during the drive?</p>

<p>A car with wheels 30 inches in diameter moves forward 8 feet in a clockwise direction. For any point on the wheel, find the angle through which the point rotates. Express your answer in radians and degrees. (note 1 foot = 12 inches)</p>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
300^{\circ}
</code></li>
</ul>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>c) <img src="/qimages/489" /></p>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>a) <img src="/qimages/487" /></p>

<p>The pendulum of a clock is 20 inches long and swing through an arc of <code class='latex inline'>20^o</code> each second. How far does the tip of the pendulum move in <code class='latex inline'>1</code> second? </p>

<p>Convert to radian measure.</p><p>60<code class='latex inline'>^{\circ}</code></p>

<p>Convert to radian measure.</p><p> <code class='latex inline'>-180^{\circ}</code></p>

<p>Determine the approximate degree measure, to the nearest tenth, for each angle.</p>
<ul>
<li>3.14</li>
</ul>

<p>Determine the exact degree measure for each angle</p>
<ul>
<li><code class='latex inline'>
\displaystyle
\frac{\pi}{9}
</code></li>
</ul>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
22.5^{\circ}
</code></li>
</ul>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>f) <img src="/qimages/492" /></p>

<p>Convert to degree measure.</p><p><code class='latex inline'>-\displaystyle{\frac{5\pi}{3}}</code></p>

<p>A particle moves <code class='latex inline'>40</code> m on a circle in <code class='latex inline'>5</code> s and the radius of the circle is <code class='latex inline'>8</code> m, determine the angular velocity of the particle in radians per second.</p>

<p>The London Eye is a large Ferris wheel located on the banks of the Tames River in London, England. Each sealed and air-conditioned passenger capsule holds about 25 passengers. The diameter of the wheel is <code class='latex inline'>135</code> m, and the wheel takes about half an hour to complete one revolution.</p><p>How long would it take a passenger to travel <code class='latex inline'>2</code> radians?</p>

<p> Find the area of the shaded region in the figure.</p><img src="/qimages/281" />

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{6}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
15^{\circ}
</code></li>
</ul>

<p>A circle of radius 25 cm has a central angle of 4.75 radians. Determine the length of the arc the subtends this angle.</p>

<p>David made a swing for his niece Sarah using ropes 2.4 m long, so that Sarah swings through an arc of length 1.2 m. Determine the angle through which Sarah swings, in both radians and degrees.</p>

<p>The measure of one of the dual angles in an isosceles triangle is twice the measure of the remaining angle. Determine the exact radian measures of the three angles in the triangle.</p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{6}</code> rad.</p><p> <code class='latex inline'>
\displaystyle
5^{\circ}
</code></p>

<p>Sketch each rotation about a circle of radius 1.</p><p> <code class='latex inline'>-\displaystyle{\frac{\pi}{4}}</code></p>

<p>Sketch each rotation about a circle of radius 1.</p><p><code class='latex inline'>\displaystyle{\frac{5\pi}{3}}</code></p>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>b) <img src="/qimages/488" /></p>

<p> Find the radian measure of the angle with the given degree measure.</p><p><code class='latex inline'>
\displaystyle
72^{\circ}
</code></p>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
3^{\circ}
</code></li>
</ul>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{6}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
150^{\circ}
</code></li>
</ul>

<p> The measures of two angles in standard position are given. Determine whether the angles are co-terminal(in the same position in the circle).</p><p><code class='latex inline'>
\displaystyle
\frac{32\pi}{3}, \frac{11\pi}{3}
</code></p>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>30^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{6}</code> rad.</p><p><code class='latex inline'>
\displaystyle
120^{\circ}
</code></p>

<p>The members of a high-school basketball team are driving from Calgary to Vancouver, which is a distance of <code class='latex inline'>675</code> km. Each tire on their van has a radius of <code class='latex inline'>32</code> cm. If the team members drive at a constant speed and cover the distance from Calgary to Vancouver in <code class='latex inline'>6</code> h <code class='latex inline'>45</code> min, what is the angular velocity, in radians/second, of each tire.</p>

<p>Convert to radian measure.</p><p><code class='latex inline'>-135^{\circ}</code></p>

<p>The central angle of a circle a measure of 3 rad. If the length of the radius is 2 cm, find the length of the arc.</p>

<p>Determine the exact radian measure for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
10^{\circ}
</code></li>
</ul>

<p>Determine the approximate radian measure, to the nearest hundredth, for each angle.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
82^{\circ}
</code></li>
</ul>

<p>Determine mentally the exact radian measure for each angle, given that <code class='latex inline'>45^{\circ}</code> is exactly <code class='latex inline'>\frac{\pi}{4}</code> rad.</p>
<ul>
<li><code class='latex inline'>
\displaystyle
180^{\circ}
</code></li>
</ul>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>d) <img src="/qimages/490" /></p>

<p>If a circle has a radius of 65 m, determine the arc length for each of the following central angles.</p><p> 150<code class='latex inline'>^{\circ}</code></p>

<p> Find the degree measure of the angle with the given radian measure.</p><p><code class='latex inline'>
\displaystyle
-1.2
</code></p>

<p>A point is rotated about a circle of radius 1. Its start and finish are
shown. State the rotation in radian measure and in degree measure.</p><p>e) <img src="/qimages/491" /></p>

<p>Determine the exact radian measure for each angle.</p><p><code class='latex inline'>
\displaystyle
15^{\circ}
</code></p>

<p>Determine the arc length of a circle with a radius of 8 cm if
the central angle is <code class='latex inline'>300^{\circ}</code></p>