5.2 Unit Circle: Sine and Cosine Functions

Find cosine and sine of the angle $\pi .$

If $\cos \left(t\right) = \frac{24}{25}$ and $t$ is in the fourth quadrant, find $\text{sin} \left(t\right) .$

Evaluate $\cos \left(\frac{5 \pi}{3}\right)$ using a graphing calculator or computer.

Evaluate $\sin \left(\frac{\pi}{3}\right) .$

Find the reference angle of $\frac{5 \pi}{3} .$

1. ⓐ Using a reference angle, find the exact value of $\cos \left(150°\right)$ and $\text{sin} \left(150°\right) .$
2. ⓑ Using the reference angle, find $\cos \frac{5 \pi}{4}$ and $\sin \frac{5 \pi}{4} .$

1. ⓐ Use the reference angle of $315°$ to find $\cos \left(315°\right)$ and $\sin \left(315°\right) .$
2. ⓑ Use the reference angle of $- \frac{\pi}{6}$ to find $\cos \left(- \frac{\pi}{6}\right)$ and $\sin \left(- \frac{\pi}{6}\right) .$

Find the coordinates of the point on the unit circle at an angle of $\frac{7 \pi}{6} .$

Find the coordinates of the point on the unit circle at an angle of $\frac{5 \pi}{3} .$

Describe the unit circle.

Discuss the difference between a coterminal angle and a reference angle.

Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

Use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by $t$ lies.

$\text{sin} \left(t\right) > 0$ and $\cos \left(t\right) > 0$

Use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by $t$ lies.

$\sin \left(t\right) < 0$ and $\cos \left(t\right) > 0$

Find the exact value of each trigonometric function.

$\sin \frac{\pi}{3}$

Find the exact value of each trigonometric function.

$\cos \frac{\pi}{3}$

Find the exact value of each trigonometric function.

$\cos \frac{\pi}{4}$

Find the exact value of each trigonometric function.

$\sin \pi$

Find the exact value of each trigonometric function.

$\cos \pi$

Find the exact value of each trigonometric function.

$\cos \frac{\pi}{6}$

State the reference angle for the given angle.

$240°$

State the reference angle for the given angle.

$100°$

State the reference angle for the given angle.

$135°$

State the reference angle for the given angle.

$\frac{2 \pi}{3}$

State the reference angle for the given angle.

$\frac{- 11 \pi}{3}$

State the reference angle for the given angle.

$\frac{- \pi}{8}$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$300°$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$135°$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$120°$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$150°$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{7 \pi}{6}$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{3 \pi}{4}$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{2 \pi}{3}$

Find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{7 \pi}{4}$

Find the requested value.

If $\text{cos} \left(t\right) = \frac{2}{9}$ and $t$ is in the 1^st quadrant, find $\text{sin} \left(t\right) .$

Find the requested value.

If $\text{sin} \left(t\right) = - \frac{1}{4}$ and $t$ is in the 3^rd quadrant, find $\text{cos} \left(t\right) .$

Find the requested value.

Find the coordinates of the point on a circle with radius 20 corresponding to an angle of $120° .$

Find the requested value.

Find the coordinates of the point on a circle with radius 16 corresponding to an angle of $\frac{5 \pi}{9} .$

Find the requested value.

State the range of the sine and cosine functions.

Use a graphing calculator to evaluate.

$\cos \frac{5 \pi}{9}$

Use a graphing calculator to evaluate.

$\cos \frac{\pi}{10}$

Use a graphing calculator to evaluate.

$\cos \frac{3 \pi}{4}$

Use a graphing calculator to evaluate.

$\cos 98°$

Use a graphing calculator to evaluate.

$\sin 310°$

Evaluate.

$\sin \left(\frac{3 \pi}{4}\right) \cos \left(\frac{5 \pi}{3}\right)$

Evaluate.

$\sin \left(\frac{- 9 \pi}{4}\right) \cos \left(\frac{- \pi}{6}\right)$

Evaluate.

$\sin \left(\frac{7 \pi}{4}\right) \cos \left(\frac{- 2 \pi}{3}\right)$

Evaluate.

$\cos \left(\frac{- \pi}{3}\right) \cos \left(\frac{\pi}{4}\right)$

Evaluate.

$\sin \left(\pi\right) \sin \left(\frac{\pi}{6}\right)$

Use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point $\left(0 , 1\right) ,$ that is, on the due north position. Assume the carousel revolves counter clockwise.

What are the coordinates of the child after 90 seconds?

Use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point $\left(0 , 1\right) ,$ that is, on the due north position. Assume the carousel revolves counter clockwise.

When will the child have coordinates $\left(0.707 , –0.707\right)$ if the ride lasts 6 minutes? (There are multiple answers.)