Question 1

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

Accepted Answer

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

Question 2

Given that cos(42°) = 0.74, find the value of sin(48°).

Accepted Answer

Step 1: Recognize that 42° and 48° are complementary angles (42° + 48° = 90°). Step 2: Apply the complementary angle rule: cos(θ) = sin(90° − θ). Step 3: Therefore, cos(42°) = sin(90° − 42°) = sin(48°). Step 4: Since cos(42°) = 0.74, we have sin(48°) = 0.74.

Question 3

If sec(37°) = 1.25, then cosec(53°) equals:

Accepted Answer

Step 1: Recognize that 37° and 53° are complementary angles (37° + 53° = 90°). Step 2: Apply the complementary angle rule: sec(θ) = cosec(90° − θ). Step 3: Therefore, sec(37°) = cosec(90° − 37°) = cosec(53°). Step 4: Since sec(37°) = 1.25, we have cosec(53°) = 1.25.

Question 4

If tan(x) = cot(40°), then the value of x is:

Accepted Answer

Step 1: Recognize that tan(x) = cot(40°). Step 2: Apply the complementary angle rule: cot(θ) = tan(90° − θ). Step 3: Therefore, cot(40°) = tan(90° − 40°) = tan(50°). Step 4: So tan(x) = tan(50°). Step 5: This means x = 50° (taking the principal value in the range 0° to 90°).

Question 5

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 6

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 7

Given that tan(28°) = m, find the value of cot(62°).

Accepted Answer

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

Question 8

If cos(42°) = 0.743, then sin(48°) is approximately equal to:

Accepted Answer

Step 1: Recognize that 42° + 48° = 90°, making them complementary angles. Step 2: Apply the complementary angle identity: cos(θ) = sin(90° − θ). Step 3: Therefore, cos(42°) = sin(90° − 42°) = sin(48°). Step 4: Since cos(42°) = 0.743, we have sin(48°) = 0.743.

Question 9

Simplify: sin(18°) × sec(72°) + cos(18°) × csc(72°)

Accepted Answer

Step 1: Recognize that 18° + 72° = 90°, so these are complementary angles. Step 2: Apply complementary identities: sec(72°) = csc(18°) and csc(72°) = sec(18°). Step 3: Substitute: sin(18°) × csc(18°) + cos(18°) × sec(18°). Step 4: Simplify using reciprocal identities: sin(18°) × (1/sin(18°)) + cos(18°) × (1/cos(18°)) = 1 + 1 = 2.

Question 10

If sin(3x) = cos(2x), where 0° < x < 30°, then the value of x is:

Accepted Answer

Step 1: Recognize that sin(θ) = cos(φ) when θ and φ are complementary, i.e., θ + φ = 90°. Step 2: Set up the equation: 3x + 2x = 90°. Step 3: Simplify: 5x = 90°. Step 4: Solve: x = 18°. Step 5: Verify: 0° < 18° < 30° ✓

Question 11

Given that sin(α) · sec(90° - α) + cos(α) · csc(90° - α) = k, find the value of k.

Accepted Answer

Step 1: Use complementary angle identities: sec(90° - α) = csc(α) and csc(90° - α) = sec(α). Step 2: Substitute: sin(α) · csc(α) + cos(α) · sec(α). Step 3: Simplify using reciprocal identities: sin(α) · (1/sin(α)) + cos(α) · (1/cos(α)) = 1 + 1 = 2. Therefore, k = 2.

Question 12

If sin(5θ + 12°) = cos(3θ + 28°), where both angles are acute, find the value of tan(2θ).

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 for θ: 8θ = 50°, so θ = 6.25°. Step 5: Calculate 2θ = 12.5°. Step 6: tan(12.5°) ≈ 0.2217, but we need exact form. Recalculate: θ = 50°/8 = 25°/4. Then 2θ = 25°/2 = 12.5°. Using tan(12.5°) = √2 - 1 (half-angle formula from tan(25°)). Alternatively, verify: sin(5×6.25° + 12°) = sin(43.25°) and cos(3×6.25° + 28°) = cos(46.75°). Since 43.25° + 46.75° = 90°, this confirms complementary angles. tan(12.5°) = √2 - 1 ≈ 0.414.

Question 13

If tan(3x - 15°) = cot(2x + 25°) and both angles are acute, find sin(x).

Accepted Answer

Step 1: Use the complementary angle property: tan(A) = cot(B) implies A + B = 90°. Step 2: Set up equation: (3x - 15°) + (2x + 25°) = 90°. Step 3: Simplify: 5x + 10° = 90°. Step 4: Solve for x: 5x = 80°, so x = 16°. Step 5: Calculate sin(16°). Using standard angle approximations or exact values: sin(16°) ≈ 0.276. However, for exact form, note that 16° is not a standard angle. Verify: tan(3×16° - 15°) = tan(33°) and cot(2×16° + 25°) = cot(57°). Since 33° + 57° = 90°, tan(33°) = cot(57°) ✓. sin(16°) is the answer, which equals approximately 0.276 or can be left as sin(16°).

Question 14

A ladder leans against a wall such that it makes an angle of 60° with the ground. If the ladder is 10 metres long, find the height at which the ladder touches the wall. Also, if sin(30°) = 1/2, what is cos(60°)?

Accepted Answer

Step 1: The ladder makes 60° with the ground, so the angle with the wall is 90° - 60° = 30°. Step 2: Height = ladder length × sin(60°) = 10 × (√3/2) = 5√3 metres. Step 3: Recognize that 60° and 30° are complementary angles (60° + 30° = 90°). Step 4: Use the complementary angle identity: cos(60°) = sin(90° - 60°) = sin(30°) = 1/2. Therefore, cos(60°) = 1/2.

SBI Clerk Complementary Angles

Complementary Angles— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Complementary Angles — Practice Questions

60-Second Revision — Complementary Angles