Reciprocal Identities Unit Circle & Domain Quiz

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

So, the final answer is sin²θ

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.

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.

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.

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.