Name the reference angle for each of the following angles.

1. \(30^\circ\)

2. \(135^\circ\)

3. \(240^\circ\)

4. \(330^\circ\)

5. \(-210^\circ\)

6. \(-140^\circ\)

Find the exact value of \(\sin 240^\circ\).

Find the exact value of \(\tan 315^\circ\)

Find the exact value of \(\csc 300^\circ\)

Find the exact value of \(\cos 495^\circ\)

Find \(\theta\) if \(\sin\theta=-0.5592\) and \(\theta\) terminates in QIII with \(0^\circ<\theta<360^\circ\).

Find \(\theta\) to the nearest tenth of a degree if \(\tan\theta=-0.8541\) and \(\theta\) terminates in QIV with \(0^\circ<\theta<360^\circ\).

Find \(\theta\) if \(\sin\theta=-\displaystyle\frac{1}{2}\) and \(\theta\) terminates in QIII with \(0^\circ<\theta<360^\circ\).

Find \(\theta\) to the nearest degree if \(\sec\theta=3.8637\) and \(\theta\) terminates in QIV with \(0^\circ<\theta<360^\circ\).

Find \(\theta\) to the nearest degree if \(\cot\theta=-1.6003\) and \(\theta\) terminates in QII with \(0^\circ<\theta<360^\circ\).

Mini Lecture

1. Draw \(120^\circ\) and name the reference angle.

2. Find the exact value of \(\cos 225^\circ\).

3. Find \(\theta\) if \(\sin\theta=-0.3090\) and \(\theta\in\) QIII.

4. Find \(\theta\) if \(\sin\theta=-\displaystyle\frac{\sqrt{3}}{2}\) and \(\theta\in\) QIII.

A central angle \(\theta\) in a circle of radius \(3\) centimeters cuts off an arc of length \(6\) centimeters. What is the measure of \(\theta\)?

Convert \(45^\circ\) to radians.

Convert \(450^\circ\) to radians.

Convert \(\displaystyle\frac{\pi}{6}\) radians to degrees.

Convert \(\displaystyle\frac{4\pi}{3}\) radians to degrees.

Find \(\sin{\displaystyle\frac{\pi}{6}}\).

Find \(4\sin{\displaystyle\frac{7\pi}{6}}\).

Evaluate \(4\sin(2x+\pi)\) when \(x=\displaystyle\frac{\pi}{6}\).

Mini Lecture

1. \(\theta\) is a central angle in a circle of radius \(r\). Find \(\theta\) if \(r=3\) and \(s=9\) cm.

2. Convert \(\displaystyle\frac{2\pi}{3}\) radians to degrees.

3. Convert \(-150^\circ\) to radians.

4. Simplify \(\sin\left(x+\displaystyle\frac{\pi}{2}\right)\) if \(x=\displaystyle\frac{\pi}{6}\)

Use the unit circle in Figure 5 to find the six trigonometric functions of \(\displaystyle\frac{5\pi}{6}\).

Use the unit circle to find all values of \(t\) between \(0\) and \(2\pi\) for which \(\cos t=\displaystyle\frac{1}{2}\).

Show that cosecant is an odd function.

Mini Lecture

1. Find all trig functions for \(\displaystyle\frac{11\pi}{6}\).

2. Find all \(\theta\) for which \(\cos\theta=-\displaystyle\frac{\sqrt{3}}{2}\).

3. Find \(\cos\left(-\displaystyle\frac{5\pi}{6}\right)\)

Enrichment
There are actually three definitions for the trigonometric functions. Some of you may have used the right triangle definition previously. This video shows how all three give equivalent results.

Give the length of the arc cut off by a central angle of \(2\) radians in a circle of radius \(4.3\) inches.

The minute hand of a clock is \(1.2\) cm long. To two significant digits, how far does the tip of the minute hand move in \(20\) minutes?

The diameter of the Ferris wheel in Figure 3 is \(250\) feet, and \(\theta\) is the central angle formed as a rider travels his or her initial position \(P_0\) to position \(P_1\). Find the distance traveled by the rider if \(\theta=45^\circ\) and if \(\theta=105^\circ\).

A person standing on the Earth notices that a \(747\) Jumbo Jet flying overhead subtends an angle of \(0.45^\circ\). If the length of the jet is \(230\) feet, find its altitude to the nearest thousand feet.

Find the area of the sector formed by a central angle of \(1.4\) radians in a circle of radius \(2.1\) meters.

If the sector formed by a central angle of \(15^\circ\) has an area of \(\displaystyle\frac{\pi}{3}\) square centimeters, find the radius of the circle.

A lawn sprinkler located at the corner of a yard is set to rotate through \(90^\circ\) and project water out \(30.0\) feet. To three significant digits, what area of the lawn is watered by the sprinkler?

Mini Lecture

1. Find the arc length of \(s\), of a circle with central angle \(\theta=60^\circ\) and radius \(r=4\) millimeters.

2. Find the radius \(r\) of a circle with central angle \(\theta=\displaystyle\frac{\pi}{4}\) and arc length \(s=\pi\) centimeters.

3. The minute hand of a clock is \(2.4\) centimeters long. How far does the tip of the minute hand travel in \(20\) minutes?

4. Find the area \(A\) of the sector formed by central angle \(\theta=15^\circ\) and \(r=5\) meters.

A point on a circle travels \(5\) centimeters in \(2\) seconds. Find the linear velocity of the point.

A point on a circle rotates through \(\displaystyle\frac{3\pi}{4}\) radians in \(3\) seconds. Give the angular velocity of \(P\).

A bicycle wheel with radius of \(13.0\) inches turns with an angular velocity of \(3\) radians per second. Find the distance traveled by a point on the bicycle tire in \(1\) minute.

Figure 1 shows a fire truck parked on the shoulder of a freeway next to a long block wall. The red light on the top of the truck is \(10\) feet from the wall and rotates through one complete revolution every \(2\) seconds. Find the equation that gives the lengths \(d\) and \(s\) in terms of time \(t\).

A phonograph record is turning at \(45\) revolutions per minute (rpm). If the distance from the center of the record to a point on the edge of the record is \(3\) inches, find the angular velocity and the linear velocity of the point in feet per minute.

Mini Lecture

1. Find the linear velocity \(v\) of a point moving with uniform circular motion, if the point covers a distance \(s=12\) centimeters in time \(t=4\) seconds.

2. Find the distance \(s\) covered by a point moving with velocity \(v=20\) feet per second for a time \(t=4\) seconds.

3. Find the distance \(s\) traveled by a point with angular velocity \(\omega=4\) radians per second on a circle of radius \(r=2\) inches, for time \(t=5\) seconds.

4. find the angular velocity \(\omega\), associated with \(10\) rpm.

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