Quotient Identities: Radians & Unit Circle Quiz

  • Grade 12th
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 14 | Questions: 20 | Updated: Jan 22, 2026
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1) The unit-circle definition implies which identity for all real θ?

Explanation

Given: unit circle x² + y² = 1. Goal: identity.

Step 1: x = cos θ, y = sin θ ⇒ cos²θ + sin²θ = 1.

So, the final answer is sin²θ + cos²θ = 1.

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About This Quiz
Quotient Identities: Radians & Unit Circle Quiz - Quiz

Connect the algebra to the circle. Convert degrees↔radians, read (cos θ, sin θ) from the unit circle, and use quadrant signs to evaluate sin, cos, and tan. By the end, your quotient identities will feel as natural as reading coordinates.

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2) The graph of y = cos θ over [0, 2π] has its maximum value at which angle?

Explanation

Given: cosine on one full period. Goal: where cos θ = 1.

Step 1: Maximum occurs at θ = 0 (also at 2π).

So, the final answer is θ = 0.

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3) If an object moves along a circle of radius 4 with central angle θ = 5π/6, what is the arc length?

Explanation

Given: r = 4, θ = 5π/6. Goal: s.

Step 1: s = rθ = 4·(5π/6) = 20π/6 = 10π/3.

So, the final answer is 10π/3.

Submit

4) On the unit circle, the angle θ has x-coordinate 0. Which could be θ?

Explanation

Given: x = cos θ = 0. Goal: choose an angle.

Step 1: cos θ = 0 at θ = π/2 and 3π/2.

Step 2: From options, π/2 fits.

So, the final answer is π/2.

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5) If cos θ = −1/2 and θ is in Quadrant III, then sin θ equals:

Explanation

Given: cos = −1/2 in QIII. Goal: sin.

Step 1: Reference angle π/3; in QIII both sin and cos are negative.

Step 2: |sin| = √3/2 ⇒ sin = −√3/2.

So, the final answer is −√3/2.

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6) Which is the correct degree-to-radian conversion factor?

Explanation

Given: deg→rad conversion. Goal: factor.

Step 1: radians = degrees × (π/180).

So, the final answer is Multiply degrees by π/180.

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7) A graph of y = sin θ on [0, 2π]. First crossing of the θ-axis after θ = 0?

Explanation

Given: zeros of sine. Goal: first positive zero after 0.

Step 1: Zeros at 0, π, 2π.

Step 2: First after 0 is π.

So, the final answer is θ = π.

Submit

8) The arc length on a unit circle equals the angle (radians) because:

Explanation

Given: s = rθ. Goal: reason on unit circle.

Step 1: With r = 1, s = θ.

So, the final answer is The radius is 1, so s = rθ = θ.

Submit

9) Which angle places the terminal side on the negative y-axis?

Explanation

Given: negative y-axis. Goal: angle.

Step 1: Coordinates (0, −1) occur at θ = 3π/2.

So, the final answer is 3π/2.

Submit

10) If θ increases from 0 to π, how does cos θ change?

Explanation

Given: cosine behavior on [0, π]. Goal: trend.

Step 1: cos 0 = 1; cos π = −1; cosine decreases on that interval.

So, the final answer is Decreases from 1 to −1.

Submit

11) Convert 225° to radians.

Explanation

Given: 225°. Goal: convert to radians.

Step 1: 225·(π/180) = (5/4)π.

So, the final answer is 5π/4.

Submit

12) Let P(cos θ, sin θ) with θ = −π/3. What are the coordinates of P?

Explanation

Given: θ = −π/3. Goal: compute cos and sin.

Step 1: cos(−π/3) = 1/2; sin(−π/3) = −√3/2.

So, the final answer is (1/2, −√3/2).

Submit

13) Convert 7π/3 radians to degrees.

Explanation

Given: 7π/3. Goal: convert to degrees.

Step 1: (7π/3)·(180/π) = 420°.

So, the final answer is 420°.

Submit

14) If the terminal point of θ is in Quadrant II, which is always true?

Explanation

Given: Quadrant II. Goal: sign pattern.

Step 1: In QII, cos < 0 and sin > 0.

So, the final answer is cos θ < 0, sin θ > 0.

Submit

15) A unit-circle point has y = −√2/2. Which angle in [0, 2π) fits?

Explanation

Given: sin θ = −√2/2. Goal: angle in [0, 2π).

Step 1: Solutions are 5π/4 and 7π/4.

Step 2: From options, 7π/4 appears.

So, the final answer is 7π/4.

Submit

16) An angle measures 1.5 radians. What is this angle in degrees?

Explanation

Given: 1.5 rad. Goal: convert to degrees.

Step 1: degrees = 1.5·(180/π) ≈ 85.94°.

So, the final answer is 85.94°.

Submit

17) Arc length on a circle of radius 6 is 3π. What is the central angle (radians)?

Explanation

Given: s = 3π, r = 6. Goal: find θ.

Step 1: s = rθ ⇒ θ = s/r = (3π)/6 = π/2.

So, the final answer is π/2.

Submit

18) If θ = 2π/3, find sin θ.

Explanation

Given: θ = 120°. Goal: sin θ.

Step 1: Reference π/3; Quadrant II makes sine positive.

Step 2: sin(2π/3) = √3/2.

So, the final answer is √3/2.

Submit

19) Which radian measure is coterminal with −π/6?

Explanation

Given: −π/6. Goal: add 2πk to get an angle in [0, 2π).

Step 1: −π/6 + 2π = 11π/6.

So, the final answer is 11π/6.

Submit

20) On the unit circle, the point for angle θ is (x, y). Which is true?

Explanation

Given: unit-circle definitions. Goal: coordinate mapping.

Step 1: Coordinates are (cos θ, sin θ).

So, the final answer is x = cos θ, y = sin θ.

Submit
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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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The unit-circle definition implies which identity for all real...
The graph of y = cos θ over [0, 2π] has its maximum value at...
If an object moves along a circle of radius 4 with central angle θ =...
On the unit circle, the angle θ has x-coordinate 0. Which could be...
If cos θ = −1/2 and θ is in Quadrant III, then sin...
Which is the correct degree-to-radian conversion factor?
A graph of y = sin θ on [0, 2π]. First crossing of the...
The arc length on a unit circle equals the angle (radians) because:
Which angle places the terminal side on the negative y-axis?
If θ increases from 0 to π, how does cos θ change?
Convert 225° to radians.
Let P(cos θ, sin θ) with θ = −π/3. What are the coordinates of...
Convert 7π/3 radians to degrees.
If the terminal point of θ is in Quadrant II, which is always...
A unit-circle point has y = −√2/2. Which angle in [0,...
An angle measures 1.5 radians. What is this angle in degrees?
Arc length on a circle of radius 6 is 3π. What is the central angle...
If θ = 2π/3, find sin θ.
Which radian measure is coterminal with −π/6?
On the unit circle, the point for angle θ is (x, y). Which is true?
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