Question 1

If sin(35°) = k, then cos(55°) = ?

Accepted Answer

Step 1: Recognize that 35° and 55° are complementary angles (35° + 55° = 90°). Step 2: Apply the complementary angle rule: sin(θ) = cos(90° − θ). Step 3: Therefore, sin(35°) = cos(55°). Step 4: Since sin(35°) = k, we have cos(55°) = k.

Question 2

If sin(35°) = k, then what is the value of cos(55°)?

Accepted Answer

Step 1: Recognize that 35° and 55° are complementary angles (35° + 55° = 90°). Step 2: Apply the complementary angle property: sin(θ) = cos(90° − θ). Step 3: Therefore, sin(35°) = cos(90° − 35°) = cos(55°). Step 4: Since sin(35°) = k, we have cos(55°) = k.

Question 3

Evaluate: [sin²(23°) + sin²(67°)] / [cos²(23°) + cos²(67°)]

Accepted Answer

Step 1: Recognize that 23° and 67° are complementary angles (23° + 67° = 90°). Step 2: Use complementary angle properties: sin(67°) = cos(23°) and cos(67°) = sin(23°). Step 3: Substitute into the numerator: sin²(23°) + sin²(67°) = sin²(23°) + cos²(23°) = 1. Step 4: Substitute into the denominator: cos²(23°) + cos²(67°) = cos²(23°) + sin²(23°) = 1. Step 5: Therefore, the expression equals 1/1 = 1.

Question 4

If sin(3x) = cos(2x), where 0° < x < 90°, find the value of x.

Accepted Answer

Step 1: Use the complementary angle property: sin(θ) = cos(90° − θ). Step 2: Rewrite the equation: sin(3x) = cos(2x) becomes sin(3x) = sin(90° − 2x). Step 3: For angles in the given range, 3x = 90° − 2x. Step 4: Solve: 3x + 2x = 90°, so 5x = 90°. Step 5: Therefore, x = 18°.

Question 5

Simplify: sin(40°)·sec(50°) + cos(40°)·csc(50°)

Accepted Answer

Step 1: Use complementary angle properties. Since 40° + 50° = 90°, we have sin(40°) = cos(50°) and cos(40°) = sin(50°). Step 2: Rewrite sec(50°) = 1/cos(50°) and csc(50°) = 1/sin(50°). Step 3: Substitute: sin(40°)·sec(50°) = cos(50°)·(1/cos(50°)) = 1. Step 4: cos(40°)·csc(50°) = sin(50°)·(1/sin(50°)) = 1. Step 5: Total = 1 + 1 = 2.

Question 6

If tan(5α) = cot(4α) and 0° < α < 18°, find α.

Accepted Answer

Step 1: Use the complementary angle property: tan(θ) = cot(90° − θ), which means cot(β) = tan(90° − β). Step 2: Rewrite cot(4α) as tan(90° − 4α). Step 3: The equation becomes: tan(5α) = tan(90° − 4α). Step 4: For angles in the given range: 5α = 90° − 4α. Step 5: Solve: 5α + 4α = 90°, so 9α = 90°. Step 6: Therefore, α = 10°.

Question 7

Given tan(62°) = m, express cot(28°) in terms of m.

Accepted Answer

Step 1: Identify that 62° and 28° are complementary angles (62° + 28° = 90°). Step 2: Apply the complementary angle property: cot(θ) = tan(90° − θ). Step 3: Therefore, cot(28°) = tan(90° − 28°) = tan(62°). Step 4: Since tan(62°) = m, we have cot(28°) = m.

Question 8

If sin²(β) + sin²(90° - β) = k, find the value of k.

Accepted Answer

Step 1: Use the complementary angle property: sin(90° - β) = cos(β). Step 2: Substitute: sin²(β) + cos²(β) = k. Step 3: Apply the Pythagorean identity: sin²(β) + cos²(β) = 1. Step 4: Therefore, k = 1.

Question 9

If sec(θ) = cosec(θ + 20°) where θ is acute, find the value of sin(2θ + 10°).

Accepted Answer

Step 1: Use the complementary angle property: sec(θ) = cosec(90° - θ). Step 2: Given sec(θ) = cosec(θ + 20°), we have cosec(90° - θ) = cosec(θ + 20°). Step 3: Since cosec is one-to-one for acute angles: 90° - θ = θ + 20°. Step 4: Solve: 90° - 20° = 2θ, so 70° = 2θ, giving θ = 35°. Step 5: Calculate 2θ + 10° = 70° + 10° = 80°. Step 6: sin(80°) = sin(90° - 10°) = cos(10°) ≈ 0.9848. However, for exact SSC answer, sin(80°) = cos(10°), which is typically left as sin(80°) or recognized as approximately 0.985. If the options require a standard value, note sin(80°) is close to 1 but not exactly a standard angle. The answer is sin(80°) = cos(10°).

Question 10

Given that tan(α) · tan(90° - α) · tan(2α) = √3, where 0° < α < 45°, find α.

Accepted Answer

Step 1: Use complementary angle property: tan(90° - α) = cot(α) = 1/tan(α). Step 2: Substitute: tan(α) · (1/tan(α)) · tan(2α) = √3. Step 3: Simplify: tan(2α) = √3. Step 4: Since tan(60°) = √3, we have 2α = 60°, so α = 30°. Step 5: Verify: 0° < 30° < 45° ✓.

SSC CPO Complementary Angles

Complementary Angles— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Complementary Angles — Practice Questions

60-Second Revision — Complementary Angles