Reciprocal Identities Unit Circle & Domain Quiz

12th Grade

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Quizzes Created: 8156 | Total Attempts: 9,588,805

| Questions: 20 | Updated: Jan 22, 2026

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1) 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

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.

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2) 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

3) 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

4) 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

5) 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

6) 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

7) 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

8) 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

9) 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

10) 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

11) 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

12) 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

13) 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

14) 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

15) 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

16) 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

17) 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

18) 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

19) 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

20) 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