Exercises: Extend trigonometric functions using unit circle - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

In a right triangle with hypotenuse 1, if $\theta$ is the angle at the origin, which of the following correctly describes $\cos(\theta)$ using right-triangle trigonometry?

A.

opposite ÷ hypotenuse

B.

adjacent ÷ hypotenuse

C.

opposite ÷ adjacent

D.

hypotenuse ÷ adjacent

2.

Which radian measure is equivalent to 270°?

A.

$\frac{\pi}{2}$

B.

$\pi$

C.

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

D.

$2\pi$

3.

A point $P$ has coordinates $(-3, 4)$ in the coordinate plane. In which quadrant is $P$ located?

A.

Quadrant I

B.

Quadrant II

C.

Quadrant III

D.

Quadrant IV

B

Fluency Practice

Evaluate each expression using the unit circle. Give exact values.

$Unit circle showing point P at (sqrt(3)/2, 1/2) in the first quadrant, with x-coordinate labeled cos(theta) and y-coordinate labeled sin(theta).$

1.

The terminal side of angle $\theta$ in standard position intersects the unit circle at the point $\left(\frac{\sqrt{3}}{2},\, \frac{1}{2}\right)$ . What is $\sin(\theta)$ ?

2.

The angle $\frac{2\pi}{3}$ has its terminal side in Quadrant II. What are the signs of $\sin\!\left(\frac{2\pi}{3}\right)$ and $\cos\!\left(\frac{2\pi}{3}\right)$ ?

A.

Both positive

B.

$\sin > 0$ , $\cos < 0$

C.

Both negative

D.

$\sin < 0$ , $\cos > 0$

3.

Use the reference angle strategy to evaluate $\sin\!\left(\frac{5\pi}{6}\right)$ . Give an exact answer.

4.

Use the reference angle strategy to evaluate $\cos\!\left(\frac{7\pi}{6}\right)$ . Give an exact answer.

5.

Evaluate $\cos\!\left(-\frac{\pi}{3}\right)$ . Give an exact answer.

C

Varied Practice

These problems use the same skills in different ways.

$Unit circle showing the sign of (cos, sin) in each quadrant: QI (+,+), QII (-,+), QIII (-,-), QIV (+,-).$

1.

In which quadrant(s) is $\sin(\theta) > 0$ and $\cos(\theta) < 0$ simultaneously?

A.

Quadrant I only

B.

Quadrant II only

C.

Quadrants II and III

D.

Quadrant IV only

2.

The angle $\frac{3\pi}{4}$ is in Quadrant ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . Its reference angle is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . Therefore $\sin\!\left(\frac{3\pi}{4}\right) = \underline{\hspace{5em}}$ and $\cos\!\left(\frac{3\pi}{4}\right) = \underline{\hspace{5em}}$ .

quadrant number:

reference angle (in terms of pi):

sin(3pi/4):

cos(3pi/4):

$Unit circle showing angle 7pi/6 in QIII with reference angle pi/6 marked between the terminal side and the negative x-axis.$

3.

The diagram shows the angle $\frac{7\pi}{6}$ and its reference angle. Use the reference angle to evaluate $\sin\!\left(\frac{7\pi}{6}\right)$ .

4.

Which of the following correctly evaluates $\sin\!\left(\frac{9\pi}{2}\right)$ ?

A.

The expression is undefined because $\frac{9\pi}{2} > 2\pi$ .

B.

$0$

C.

$1$

D.

$-1$

5.

What is the reference angle (in radians) for $\theta = \frac{5\pi}{3}$ ? Give your answer as a fraction involving $\pi$ .

D

Word Problems

Read each problem carefully. Use the unit circle to find exact values.

1.

A Ferris wheel has a radius of 1 meter and its center at the origin. A car starts at the rightmost point $(1, 0)$ and travels counterclockwise. When the car has traveled an arc length equal to $\frac{5\pi}{4}$ meters, what is the car's height above or below the center? (A positive answer means above the center.)

Find the car's height, which equals $\sin\!\left(\frac{5\pi}{4}\right)$ . Give an exact answer.

$Unit circle showing that -pi/4 (clockwise rotation) and 7pi/4 (counterclockwise) land on the same point in QIV at (sqrt(2)/2, -sqrt(2)/2).$

2.

A clock pendulum swings past the lowest point and is modeled on the unit circle. The angle $-\frac{\pi}{4}$ represents a clockwise swing of $\frac{\pi}{4}$ radians from the starting position at $(1, 0)$ .

What is $\cos\!\left(-\frac{\pi}{4}\right)$ ? Give an exact answer.

3.

A particle travels counterclockwise around the unit circle starting at $(1, 0)$ . After traveling an arc of $\frac{4\pi}{3}$ radians, the particle is at a new position.

1.

What is the $y$ -coordinate of the particle's position? This equals $\sin\!\left(\frac{4\pi}{3}\right)$ . Give an exact answer.

2.

What is the $x$ -coordinate of the particle's position? This equals $\cos\!\left(\frac{4\pi}{3}\right)$ . Give an exact answer.

E

Error Analysis

Each problem shows a student's work that contains an error. Identify the mistake.

1.

Jordan evaluated $\cos\!\left(\frac{2\pi}{3}\right)$ as follows:

"The angle $\frac{2\pi}{3}$ is in Quadrant II. The adjacent side is always positive because it is a length. So $\cos\!\left(\frac{2\pi}{3}\right) = \frac{1}{2}$ ."

What is the error in Jordan's reasoning?

A.

Jordan used the wrong reference angle.

B.

Jordan confused sine and cosine.

C.

Jordan treated cosine as a distance (always positive) instead of an x-coordinate (signed).

D.

Jordan applied the reference angle formula for the wrong quadrant.

2.

Alex evaluated $\sin\!\left(-\frac{5\pi}{6}\right)$ and $\cos\!\left(-\frac{5\pi}{6}\right)$ :

"The angle is negative, so both sine and cosine must be negative. $\sin\!\left(-\frac{5\pi}{6}\right) = -\frac{1}{2}$ and $\cos\!\left(-\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ ."

Alex's final numerical values are actually correct, but the reasoning is wrong. Which statement best explains why?

A.

A negative angle always produces negative values for both sine and cosine.

B.

The signs of sine and cosine depend on the quadrant of the terminal side, not on whether the angle itself is negative.

C.

Negative angles are undefined on the unit circle.

D.

The reference angle for negative angles is always positive, so the values must be positive.

F

Challenge / Extension

These problems require multiple steps or deeper reasoning. Try them if you have mastered the core problems.

1.

Evaluate $\cos\!\left(\frac{17\pi}{6}\right)$ . Show the steps: (1) find a coterminal angle in $[0, 2\pi)$ , (2) identify the quadrant and reference angle, (3) attach the correct sign.

2.