Question 1

Evaluate: sin²(18°) + sin²(72°)

Accepted Answer

Step 1: Recognize that 18° and 72° are complementary angles (18° + 72° = 90°). Step 2: Apply the complementary angle rule: sin(72°) = sin(90° - 18°) = cos(18°). Step 3: Substitute: sin²(18°) + cos²(18°). Step 4: Apply the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 5: Therefore, sin²(18°) + sin²(72°) = 1.

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 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 3

Evaluate: tan(28°) × tan(62°)

Accepted Answer

Step 1: Recognize that 28° and 62° are complementary angles (28° + 62° = 90°). Step 2: Apply the complementary angle rule: tan(θ) × tan(90° - θ) = 1. Step 3: Therefore, tan(28°) × tan(62°) = tan(28°) × cot(28°) = 1.

Question 4

If cos(42°) = 0.743, what is sin(48°)?

Accepted Answer

Step 1: Recognize that 42° and 48° are complementary angles (42° + 48° = 90°). Step 2: Apply the complementary angle rule: sin(θ) = cos(90° - θ). Step 3: Therefore, sin(48°) = cos(90° - 48°) = cos(42°). Step 4: Since cos(42°) = 0.743, we have sin(48°) = 0.743.

Question 5

Simplify: sec(37°) × sin(53°) - 1

Accepted Answer

Step 1: Recognize that 37° and 53° are complementary angles (37° + 53° = 90°). Step 2: Apply sin(53°) = cos(37°) using the complementary angle rule. Step 3: Substitute: sec(37°) × cos(37°) - 1. Step 4: Since sec(37°) = 1/cos(37°), we have [1/cos(37°)] × cos(37°) - 1 = 1 - 1 = 0.

Question 6

If cot(x) = tan(25°), then what is the value of x (in degrees)?

Accepted Answer

Step 1: Recall that cot(θ) = tan(90° - θ) using the complementary angle rule. Step 2: Therefore, cot(x) = tan(90° - x). Step 3: Given cot(x) = tan(25°), we have tan(90° - x) = tan(25°). Step 4: Equating angles: 90° - x = 25°. Step 5: Solve for x: x = 90° - 25° = 65°.

Question 7

Simplify: sin(73°)·sec(17°) + cos(73°)·csc(17°)

Accepted Answer

Step 1: Recognize that 73° + 17° = 90°, so these are complementary angles. Step 2: Apply complementary identities: sin(73°) = cos(17°) and cos(73°) = sin(17°). Step 3: Rewrite the expression: cos(17°)·sec(17°) + sin(17°)·csc(17°). Step 4: Simplify using sec(θ) = 1/cos(θ) and csc(θ) = 1/sin(θ): cos(17°)·(1/cos(17°)) + sin(17°)·(1/sin(17°)) = 1 + 1 = 2.

Question 8

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

Accepted Answer

Step 1: Observe that 62° + 28° = 90°, so these are complementary angles. Step 2: Apply the complementary angle identity: 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 9

If cos(48°) = 0.67, what is the approximate value of sin(42°)?

Accepted Answer

Step 1: Recognize that 48° + 42° = 90°, making them complementary angles. Step 2: Apply the complementary angle identity: sin(θ) = cos(90° − θ). Step 3: Therefore, sin(42°) = cos(90° − 42°) = cos(48°). Step 4: Since cos(48°) = 0.67, we have sin(42°) = 0.67.

Question 10

If sin(35°) = cos(θ), where θ is an acute angle, then find the value of tan(θ) + cot(θ).

Accepted Answer

Step 1: Since sin(35°) = cos(θ), we use the complementary angle property: sin(A) = cos(90° - A). Therefore, sin(35°) = cos(90° - 35°) = cos(55°). Step 2: So θ = 55°. Step 3: Now we need to find tan(55°) + cot(55°). Step 4: tan(55°) + cot(55°) = sin(55°)/cos(55°) + cos(55°)/sin(55°) = [sin²(55°) + cos²(55°)] / [sin(55°)cos(55°)]. Step 5: Using sin²(55°) + cos²(55°) = 1, we get 1 / [sin(55°)cos(55°)]. Step 6: Using the double angle formula: 2sin(55°)cos(55°) = sin(110°) = sin(70°) ≈ 0.9397. Therefore sin(55°)cos(55°) ≈ 0.4699. Step 7: So tan(55°) + cot(55°) = 1/0.4699 ≈ 2.128 ≈ 2/sin(110°). Alternatively, using 1/[sin(55°)cos(55°)] = 2/sin(110°) = 2/sin(70°) ≈ 2.13. For exact calculation: tan(55°) + cot(55°) = 2/sin(110°) = 2csc(110°). Since sin(110°) = sin(70°) and this evaluates to approximately 2.13, the closest standard answer is 2csc(70°) or approximately 2.13. However, using complementary relationship more directly: tan(θ) + cot(θ) = 2csc(2θ) = 2csc(110°). This gives us 2/sin(110°) which equals 2/cos(20°) ≈ 2.128. The answer evaluates to approximately 2.13 or exactly 2csc(70°).

Question 11

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 12

Evaluate: tan(28°) × tan(62°)

Accepted Answer

Step 1: Observe that 28° and 62° are complementary angles (28° + 62° = 90°). Step 2: Apply the complementary angle rule: tan(θ) × tan(90° − θ) = 1. Step 3: Therefore, tan(28°) × tan(62°) = tan(28°) × cot(28°) = 1.

Question 13

If cos(42°) = m, then sin(48°) equals:

Accepted Answer

Step 1: Recognize that 42° and 48° are complementary angles (42° + 48° = 90°). 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°) = m, we have sin(48°) = m.

Question 14

Simplify: cot(67°) − tan(23°)

Accepted Answer

Step 1: Recognize that 67° and 23° are complementary angles (67° + 23° = 90°). Step 2: Apply the complementary angle identity: cot(θ) = tan(90° − θ). Step 3: Therefore, cot(67°) = tan(90° − 67°) = tan(23°). Step 4: Substitute: cot(67°) − tan(23°) = tan(23°) − tan(23°) = 0.

Question 15

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

Accepted Answer

Step 1: Observe that 62° + 28° = 90°, so these are complementary angles. Step 2: Apply the complementary angle identity for tangent and cotangent: 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 16

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

Accepted Answer

Step 1: Observe that 62° + 28° = 90°, so these are complementary angles. 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 17

If sin(3x) = cos(2x), where 0° < x < 30°, find the value of x.

Accepted Answer

Step 1: Use the complementary angle identity sin(θ) = cos(90° − θ). Step 2: Rewrite sin(3x) = cos(2x) as cos(90° − 3x) = cos(2x). Step 3: Since cosine is a one-to-one function in the given range, equate the angles: 90° − 3x = 2x. Step 4: Solve for x: 90° = 5x, so x = 18°. Step 5: Verify: 0° < 18° < 30° ✓

Question 18

Given that cos(48°) = 0.67, find the approximate value of sin(42°).

Accepted Answer

Step 1: Recognize that 48° + 42° = 90°, making them complementary angles. Step 2: Apply the complementary angle identity: sin(θ) = cos(90° − θ). Step 3: Therefore, sin(42°) = cos(90° − 42°) = cos(48°). Step 4: Since cos(48°) = 0.67, we have sin(42°) = 0.67.

Question 19

If sec(36°) = p, then what is cosec(54°)?

Accepted Answer

Step 1: Observe that 36° + 54° = 90°, so these are complementary angles. Step 2: Apply the complementary angle identity: sec(θ) = cosec(90° − θ). Step 3: Therefore, sec(36°) = cosec(90° − 36°) = cosec(54°). Step 4: Since sec(36°) = p, we have cosec(54°) = p.

Question 20

Simplify: sin(73°)cos(17°) + cos(73°)sin(17°)

Accepted Answer

Step 1: Recognize that 73° + 17° = 90°, so 17° = 90° − 73°. Step 2: Apply complementary angle identities: cos(17°) = cos(90° − 73°) = sin(73°) and sin(17°) = sin(90° − 73°) = cos(73°). Step 3: Substitute into the expression: sin(73°)·sin(73°) + cos(73°)·cos(73°) = sin²(73°) + cos²(73°). Step 4: Apply the Pythagorean identity: sin²(73°) + cos²(73°) = 1.

Question 21

If sin(5x) = cos(4x), where 0° < x < 18°, 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(5x) = cos(4x) as sin(5x) = sin(90° − 4x). Step 3: For this equality to hold: 5x = 90° − 4x. Step 4: Solve for x: 5x + 4x = 90°, so 9x = 90°, therefore x = 10°. Step 5: Verify: 0° < 10° < 18° ✓

Question 22

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 23

If cos(3x) = sin(2x + 10°), where 0° < x < 30°, find the value of x.

Accepted Answer

Step 1: Recognize that cos(θ) = sin(90° − θ) is the complementary angle identity. Step 2: Rewrite cos(3x) as sin(90° − 3x). Step 3: Set up the equation: sin(90° − 3x) = sin(2x + 10°). Step 4: For equal sines, angles must be equal (within the given range): 90° − 3x = 2x + 10°. Step 5: Solve: 90° − 10° = 2x + 3x → 80° = 5x → x = 16°.

Question 24

In a right-angled triangle, one acute angle is 37°. If sin(37°) = 0.6, what is the value of cos(53°)?

Accepted Answer

Step 1: In a right-angled triangle, the two acute angles are complementary (sum to 90°). Step 2: The two acute angles are 37° and 53° (since 37° + 53° = 90°). Step 3: Apply the complementary angle identity: sin(37°) = cos(90° − 37°) = cos(53°). Step 4: Therefore, cos(53°) = sin(37°) = 0.6.

Question 25

Evaluate: [sin(37°)/cos(53°)] + [tan(29°)/cot(61°)]

Accepted Answer

Step 1: Recognize that 37° + 53° = 90° and 29° + 61° = 90°, so these are complementary angle pairs. Step 2: Apply complementary identities: sin(37°) = cos(53°) and tan(29°) = cot(61°). Step 3: Substitute into the expression: [cos(53°)/cos(53°)] + [cot(61°)/cot(61°)]. Step 4: Simplify: 1 + 1 = 2.

Question 26

If cos(7x - 20°) = sin(3x + 50°), where 0° < x < 20°, find x.

Accepted Answer

Step 1: Use complementary angle property: cos(A) = sin(B) implies A + B = 90°. Step 2: Set up equation: (7x - 20°) + (3x + 50°) = 90°. Step 3: Simplify: 10x + 30° = 90°. Step 4: Solve: 10x = 60°. Step 5: Therefore x = 6°.

Question 27

If sin(3x − 15°) = cos(x + 35°) and both angles are in the first quadrant, what is the value of x?

Accepted Answer

Step 1: Use sin(A) = cos(B) ⟹ A + B = 90°. Step 2: Set up: (3x − 15°) + (x + 35°) = 90°. Step 3: Combine: 4x + 20° = 90°. Step 4: Solve: 4x = 70°, so x = 17.5°.

Question 28

Given that tan(2α) · cot(88° − 2α) = 1 and α is acute, find 2α.

Accepted Answer

Step 1: Use the property: tan(A) · cot(B) = 1 when A + B = 90°. Step 2: Set up: 2α + (88° − 2α) = 90°. Step 3: Simplify: 88° = 90°. This is false, so use tan(A) · cot(A) = 1 instead. Step 4: Recognize cot(88° − 2α) = tan(90° − (88° − 2α)) = tan(2° + 2α). Step 5: For tan(2α) · tan(2° + 2α) = 1, we need 2α + (2° + 2α) = 90°. Step 6: Solve: 4α + 2° = 90°, so 4α = 88°, thus 2α = 44°.

Question 29

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 30

A ladder leans against a wall such that it makes an angle θ with the ground. If sin(θ) = 3/5 and the ladder makes a complementary angle with the wall, find the value of tan(θ) + cot(90° − θ) − sec(90° − θ).

Accepted Answer

Step 1: Given sin(θ) = 3/5, find cos(θ) using sin²(θ) + cos²(θ) = 1. Step 2: (3/5)² + cos²(θ) = 1 ⟹ 9/25 + cos²(θ) = 1 ⟹ cos²(θ) = 16/25 ⟹ cos(θ) = 4/5 (positive since θ is acute). Step 3: Calculate tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4. Step 4: Use complementary angle identities: cot(90° − θ) = tan(θ) = 3/4 and sec(90° − θ) = cosec(θ) = 1/sin(θ) = 5/3. Step 5: Compute: tan(θ) + cot(90° − θ) − sec(90° − θ) = 3/4 + 3/4 − 5/3 = 6/4 − 5/3 = 3/2 − 5/3 = 9/6 − 10/6 = −1/6.

RRB NTPC Complementary Angles

RRB NTPC Complementary Angles — Past Exam Questions

Complementary Angles— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Complementary Angles — Practice Questions

60-Second Revision — Complementary Angles