Question 1

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 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°) in terms of k?

Accepted Answer

Step 1: Recognize that 35° and 55° are complementary angles (35° + 55° = 90°). Step 2: Apply the complementary angle identity: 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

If tan(62°) = m, then cot(28°) equals:

Accepted Answer

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

Question 4

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

Accepted Answer

Step 1: Use the complementary angle identity sin(θ) = cos(90° − θ). Step 2: Rewrite the equation: sin(3x) = cos(2x) becomes sin(3x) = sin(90° − 2x). Step 3: For this equality, 3x = 90° − 2x. Step 4: Solve: 3x + 2x = 90°, so 5x = 90°, giving x = 18°. Step 5: Verify: sin(54°) ≈ 0.809 and cos(36°) ≈ 0.809 ✓

Question 5

If cos(4α) = sin(α + 30°), where 0° < α < 30°, then α equals:

Accepted Answer

Step 1: Use the complementary angle identity cos(θ) = sin(90° − θ). Step 2: Rewrite: cos(4α) = sin(90° − 4α). Step 3: Set equal to the given expression: sin(90° − 4α) = sin(α + 30°). Step 4: Equate arguments: 90° − 4α = α + 30°. Step 5: Solve: 90° − 30° = α + 4α, so 60° = 5α, giving α = 12°. Step 6: Verify: cos(48°) ≈ 0.669 and sin(42°) ≈ 0.669 ✓

Question 6

Simplify: sin(25°)·sec(65°) + cos(25°)·csc(65°)

Accepted Answer

Step 1: Recognize that 25° and 65° are complementary (25° + 65° = 90°). Step 2: Apply complementary angle identities: sin(25°) = cos(65°) and cos(25°) = sin(65°). Step 3: Also, sec(65°) = 1/cos(65°) and csc(65°) = 1/sin(65°). Step 4: Substitute: sin(25°)·sec(65°) = cos(65°)·(1/cos(65°)) = 1. Step 5: And cos(25°)·csc(65°) = sin(65°)·(1/sin(65°)) = 1. Step 6: Total = 1 + 1 = 2.

Question 7

If tan(5θ) = cot(3θ + 20°), where 0° < θ < 10°, find θ.

Accepted Answer

Step 1: Use the complementary angle identity tan(A) = cot(90° − A), which means cot(B) = tan(90° − B). Step 2: Rewrite: cot(3θ + 20°) = tan(90° − (3θ + 20°)) = tan(70° − 3θ). Step 3: Set equal: tan(5θ) = tan(70° − 3θ). Step 4: Equate arguments: 5θ = 70° − 3θ. Step 5: Solve: 5θ + 3θ = 70°, so 8θ = 70°, giving θ = 8.75°. Step 6: Verify: tan(43.75°) ≈ 0.946 and cot(46.25°) ≈ 0.946 ✓

Question 8

If sin(5θ + 12°) = cos(3θ + 28°), where both angles are acute, find the value of θ.

Accepted Answer

Step 1: Use the complementary angle property: sin(A) = cos(B) implies A + B = 90°. Step 2: Set up equation: (5θ + 12°) + (3θ + 28°) = 90°. Step 3: Simplify: 8θ + 40° = 90°. Step 4: Solve: 8θ = 50°, so θ = 6.25°.

Question 9

Given that tan(2α + 15°) × cot(3α - 45°) = 1 and both angles are acute, find 5α.

Accepted Answer

Step 1: Use the property: tan(A) × cot(B) = 1 implies A = B (when both are acute). Step 2: Set up: 2α + 15° = 3α - 45°. Step 3: Rearrange: 15° + 45° = 3α - 2α. Step 4: Solve: 60° = α. Step 5: Therefore, 5α = 300°.

Question 10

If sin²(7θ) + cos²(7θ) = 1 and sin(4θ + 20°) = cos(2θ + 10°), find the value of θ such that 0° < θ < 30°.

Accepted Answer

Step 1: The first equation sin²(7θ) + cos²(7θ) = 1 is always true (Pythagorean identity); it provides no constraint. Step 2: Use the second equation: sin(4θ + 20°) = cos(2θ + 10°). Step 3: Apply complementary angle property: (4θ + 20°) + (2θ + 10°) = 90°. Step 4: Simplify: 6θ + 30° = 90°. Step 5: Solve: 6θ = 60°, so θ = 10°.

Question 11

A ladder leans against a wall such that the angle between the ladder and the ground is θ. If sin(θ) = 3/5 and the angle between the ladder and the wall is φ, find tan(φ).

Accepted Answer

Step 1: The angle between the ladder and the ground is θ. The angle between the ladder and the wall is φ. Step 2: These are complementary: θ + φ = 90°, so φ = 90° - θ. Step 3: Given sin(θ) = 3/5, find cos(θ) = 4/5 (using Pythagorean identity). Step 4: tan(φ) = tan(90° - θ) = cot(θ) = cos(θ)/sin(θ) = (4/5)/(3/5) = 4/3.

Question 12

If cos(3x - 10°) = sin(2x + 40°) and x is acute, find the value of sin(x + 15°).

Accepted Answer

Step 1: Use cos(A) = sin(B) ⟹ A + B = 90°. Step 2: (3x - 10°) + (2x + 40°) = 90°. Step 3: 5x + 30° = 90°. Step 4: 5x = 60°, so x = 12°. Step 5: sin(x + 15°) = sin(27°). Since sin(27°) = cos(63°), and using complementary angle properties, sin(27°) ≈ 0.454. For standard exam purposes, express as sin(27°) or recognize it equals cos(63°).

IBPS Clerk Complementary Angles

Complementary Angles— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Complementary Angles — Practice Questions

60-Second Revision — Complementary Angles