Reciprocal Identities Unit Circle & Domain Quiz

Grade 12th

Reviewed by Cierra Henderson

Cierra Henderson, MBA |

K-12 Expert

Review Board Member

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.

, MBA

By Thames

Thames

Community Contributor

Quizzes Created: 11121 | Total Attempts: 9,743,875

| Attempts: 13 | Questions: 20 | Updated: Jan 22, 2026

Share on Facebook Share on Twitter Share on Whatsapp Share on Pinterest Share on Email Copy to Clipboard Embed on your website

Question 1 / 21

🏆 Rank #-- ▾

0 %

0/100

Score 0/100

1) On unit circle, (cos θ, sin θ) = (−√2/2, −√2/2). Which is true?

Cot θ = −1/√2

Sec θ = √2

Csc θ = −√2

Sec θ undefined

Step 1: sin θ = −√2/2 and cos θ = −√2/2.

Step 2: csc θ = 1/sin θ = 1 / (−√2/2) = −√2.

So, the final answer is csc θ = −√2.

Explanation

Step 1: sin θ = −√2/2 and cos θ = −√2/2.

Step 2: csc θ = 1/sin θ = 1 / (−√2/2) = −√2.

So, the final answer is csc θ = −√2.

Submit

About This Quiz

Reciprocal Identities Unit Circle & Domain Quiz - Quiz

Know where functions live—and where they blow up. Use unit-circle coordinates and radian angles to decide when sec, csc, and cot are defined, evaluate exact values, and solve short equations. Master the “allowed angles” story behind the reciprocals.

2)

What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) If sin θ = 1/2 and cos θ < 0, then csc θ and sec θ are

Csc θ = 2, sec θ = 2

Csc θ = 2, sec θ = −2

Csc θ = 2, sec θ = −2/√3

Csc θ = 2, sec θ = 2/√3

Step 1: csc θ = 1/sin θ = 1 / (1/2) = 2.

Step 2: Since sin θ = 1/2, cos² θ = 1 − (1/2)² = 3/4, so cos θ = −√3/2 (because cos < 0).

Step 3: sec θ = 1/cos θ = 1 / (−√3/2) = −2/√3.

So, the final answer is csc θ = 2 and sec θ = −2/√3.

Explanation

Submit

3) Evaluate sec(−π/2).

−1

Undefined

Step 1: sec θ = 1/cos θ.

Step 2: cos(−π/2) = 0, so sec(−π/2) = 1/0.

Step 3: Division by zero is undefined.

So, the final answer is undefined.

Explanation

Step 1: sec θ = 1/cos θ.

Step 2: cos(−π/2) = 0, so sec(−π/2) = 1/0.

Step 3: Division by zero is undefined.

So, the final answer is undefined.

Submit

4) The domain of sec θ excludes

Sin θ = 0

Cos θ = 0

Tan θ = 0

Tan θ undefined

Step 1: sec θ = 1/cos θ.

Step 2: It is undefined where cos θ = 0.

Step 3: These occur at θ = π/2 and 3π/2.

So, the final answer is θ where cos θ = 0.

Explanation

Step 1: sec θ = 1/cos θ.

Step 2: It is undefined where cos θ = 0.

Step 3: These occur at θ = π/2 and 3π/2.

So, the final answer is θ where cos θ = 0.

Submit

5) If cos θ = 0 and sin θ = −1, which is true?

Sec θ = 0

Csc θ = −1

Cot θ undefined

Sec θ defined

Step 1: csc θ = 1/sin θ = 1 / (−1) = −1.

Step 2: sec θ = 1/cos θ → undefined since cos θ = 0.

So, the final answer is csc θ = −1.

Explanation

Step 1: csc θ = 1/sin θ = 1 / (−1) = −1.

Step 2: sec θ = 1/cos θ → undefined since cos θ = 0.

So, the final answer is csc θ = −1.

Submit

6) Simplify tan²θ / sec²θ

Sin²θ

Cos²θ

Tan²θ

Step 1: tan²θ / sec²θ = (sin²θ / cos ²θ) × cos²θ = sin²θ

So, the final answer is sin²θ

Explanation

Step 1: tan²θ / sec²θ = (sin²θ / cos ²θ) × cos²θ = sin²θ

So, the final answer is sin²θ

Submit

7) Simplify (1 − cosθ)/sinθ

Tan(θ/2)

Secθ − tanθ

Cscθ − cotθ

Sinθ/(1 + cosθ)

Step 1: Multiply numerator and denominator by (1 + cos θ).

Step 2: (1 − cos² θ) / [sin θ (1 + cos θ)] = sin² θ / [sin θ (1 + cos θ)] = sin θ / (1 + cos θ).

Step 3: sin θ / (1 + cos θ) = csc θ − cot θ.

So, the final answer is csc θ − cot θ.

Explanation

Submit

8) Find θ in [0, 2π) where csc θ = −1.

π/2

3π/2

Step 1: csc θ = 1/sin θ = −1 means sin θ = −1.

Step 2: sin θ = −1 at θ = 3π/2.

So, the final answer is θ = 3π/2.

Explanation

Step 1: csc θ = 1/sin θ = −1 means sin θ = −1.

Step 2: sin θ = −1 at θ = 3π/2.

So, the final answer is θ = 3π/2.

Submit

9) Evaluate cot(−π/4).

−1

Undefined

Step 1: cot(−θ) = cos(−θ)/sin(−θ) = cos θ / (−sin θ) = −cot θ.

Step 2: cot(π/4) = 1, so cot(−π/4) = −1.

So, the final answer is −1.

Explanation

Step 1: cot(−θ) = cos(−θ)/sin(−θ) = cos θ / (−sin θ) = −cot θ.

Step 2: cot(π/4) = 1, so cot(−π/4) = −1.

So, the final answer is −1.

Submit

10) Evaluate csc(−π/3).

−2/√3

2/√3

−√3/2

√3/2

Step 1: sin(−π/3) = −√3/2.

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

So, the final answer is −2/√3.

Explanation

Step 1: sin(−π/3) = −√3/2.

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

So, the final answer is −2/√3.

Submit

11) Which reciprocal identity is always true?

Tan θ = 1/cot θ

Cos θ = 1/sec θ

Sin θ = 1/csc θ

All of the above

Step 1: Recall reciprocal identities. They are sin θ = 1/csc θ, cos θ = 1/sec θ, and tan θ = 1/cot θ.

Step 2: Each relationship holds for all θ where the functions are defined.

So, the final answer is All of the above.

Explanation

Submit

12) Simplify completely: (secθ − 1)(secθ + 1)

Tan²θ

Sec²θ − 1

1 − tan²θ

Sin²θ

Step 1: Use the difference of squares: (secθ − 1)(secθ + 1) = sec² θ − 1.

Step 2: Use the Pythagorean identity sec² θ − 1 = tan² θ.

So, the final answer is tan² θ.

Explanation

Step 1: Use the difference of squares: (secθ − 1)(secθ + 1) = sec² θ − 1.

Step 2: Use the Pythagorean identity sec² θ − 1 = tan² θ.

So, the final answer is tan² θ.

Submit

13) Find θ in [0, 2π) where sin θ = 0.

π/6, 5π/6

π/2, 3π/2

π/3, 4π/3

0, π

Step 1: sin θ = 0 on the x-axis.

Step 2: On the unit circle, that occurs at θ = 0 and θ = π.

So, the final answer is θ = 0 and π.

Explanation

Step 1: sin θ = 0 on the x-axis.

Step 2: On the unit circle, that occurs at θ = 0 and θ = π.

So, the final answer is θ = 0 and π.

Submit

14) For which θ in [0, 2π) is tan θ undefined?

π/2, 3π/2

0, π

π/3, 2π/3

π/4, 5π/4

Step 1: tan θ = sin θ / cos θ.

Step 2: Tangent is undefined when cos θ = 0.

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

So, the final answer is θ = π/2 and 3π/2.

Explanation

Step 1: tan θ = sin θ / cos θ.

Step 2: Tangent is undefined when cos θ = 0.

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

So, the final answer is θ = π/2 and 3π/2.

Submit

15) If sec θ = 2, what is cos θ?

√3/2

1/2

−1/2

Step 1: sec θ = 1/cos θ.

Step 2: cos θ = 1 / sec θ = 1 / 2 = 1/2.

So, the final answer is 1/2.

Explanation

Step 1: sec θ = 1/cos θ.

Step 2: cos θ = 1 / sec θ = 1 / 2 = 1/2.

So, the final answer is 1/2.

Submit

16) Solve: sec x = 2, 0 ≤ x < 2π

0, π

π/4, 3π/4

π/2, 3π/2

π/3, 5π/3

Step 1: sec x = 2 ⇒ cos x = 1/2.

Step 2: cos x = 1/2 at x = π/3 and x = 5π/3 (Quadrants I and IV).

So, the final answer is x = π/3 and 5π/3.

Explanation

Step 1: sec x = 2 ⇒ cos x = 1/2.

Step 2: cos x = 1/2 at x = π/3 and x = 5π/3 (Quadrants I and IV).

So, the final answer is x = π/3 and 5π/3.

Submit

17) Simplify 1/(1−sinθ) − 1/(1+sinθ)

1 − sin²θ

Tanθ

2 cosθ

2 sinθ / cos²θ

Step 1: Take the common denominator (1 − sin θ)(1 + sin θ) = 1 − sin² θ = cos² θ.

Step 2: Subtract the numerators: (1 + sin θ) − (1 − sin θ) = 2 sin θ.

Step 3: Combine: (2 sin θ) / cos² θ.

So, the final answer is 2 sin θ / cos² θ.

Explanation

Submit

18) Simplify completely: 1/sin(²θ) − cot(²θ

Tan θ

Csc²θ

Sec²θ

Step 1: Rewrite each term using reciprocal identities.

1/sin(²θ) = csc(²θ), and cot(²θ) = cos(²θ)/sin(²θ).

Step 2: Combine into a single fraction:

csc(²θ) − cot(²θ) = (1 − cos(²θ)) / sin(²θ).

Step 3: Use double-angle identities.

1 − cos(²θ) = 2sin²θ, and sin(²θ) = 2sinθcosθ.

Step 4: Substitute and simplify:

(2sin²θ) / (2sinθcosθ) = sinθ / cosθ = tanθ.

So, the final answer is tan θ.

Explanation

Submit

19) Evaluate sec(π/3).

1/2

√3/2

2/√3

Step 1: sec θ = 1/cos θ.

Step 2: cos(π/3) = 1/2.

Step 3: sec(π/3) = 1 / (1/2) = 2.

So, the final answer is 2.

Explanation

Step 1: sec θ = 1/cos θ.

Step 2: cos(π/3) = 1/2.

Step 3: sec(π/3) = 1 / (1/2) = 2.

So, the final answer is 2.

Submit

20) For which θ is csc θ undefined?

θ = 2kπ

θ = π/2 + 2kπ

θ = kπ

θ = 3π/2 + 2kπ

Step 1: csc θ = 1/sin θ.

Step 2: csc θ is undefined whenever sin θ = 0.

Step 3: sin θ = 0 at θ = kπ (for any integer k).

So, the final answer is θ = kπ.

Explanation

Step 1: csc θ = 1/sin θ.

Step 2: csc θ is undefined whenever sin θ = 0.

Step 3: sin θ = 0 at θ = kπ (for any integer k).

So, the final answer is θ = kπ.

Submit