Question 1

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 2

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 3

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 4

In a right-angled triangle, if the opposite side is 5 and the hypotenuse is 13, what is the value of cos θ?

Accepted Answer

Step 1: Use the Pythagorean theorem to find the adjacent side: adjacent² + 5² = 13². Step 2: adjacent² + 25 = 169. Step 3: adjacent² = 144, so adjacent = 12. Step 4: cos θ = adjacent/hypotenuse = 12/13. Therefore, the answer is 12/13.

Question 5

In a right-angled triangle, if tan α = 12/5, what is the value of sec α?

Accepted Answer

Step 1: Use the identity 1 + tan²α = sec²α. Step 2: Substitute tan α = 12/5: 1 + (12/5)² = sec²α. Step 3: 1 + 144/25 = sec²α. Step 4: (25 + 144)/25 = sec²α. Step 5: 169/25 = sec²α. Step 6: sec α = 13/5 (positive in first quadrant). Therefore, the answer is 13/5.

Question 6

If cot β = 8/15, what is the value of sin β (where β is acute)?

Accepted Answer

Step 1: From cot β = 8/15, we have adjacent/opposite = 8/15. Step 2: In a right triangle, if adjacent = 8 and opposite = 15, then hypotenuse = √(8² + 15²) = √(64 + 225) = √289 = 17. Step 3: sin β = opposite/hypotenuse = 15/17. Therefore, the answer is 15/17.

Question 7

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 8

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 9

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 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 cot θ = 8/15 and θ is in the first quadrant, find the value of cosec θ.

Accepted Answer

Step 1: cot θ = 8/15 means adjacent/opposite = 8/15. Using the Pythagorean theorem, hypotenuse = √(8² + 15²) = √(64 + 225) = √289 = 17. Step 2: sin θ = opposite/hypotenuse = 15/17. Step 3: cosec θ = 1/sin θ = 17/15.

Question 12

If sec θ − tan θ = 1/3, find the value of sec θ + tan θ.

Accepted Answer

Step 1: Use the identity sec²θ − tan²θ = 1, which factors as (sec θ − tan θ)(sec θ + tan θ) = 1. Step 2: Substitute sec θ − tan θ = 1/3: (1/3)(sec θ + tan θ) = 1. Step 3: sec θ + tan θ = 3.

Question 13

A ladder 20 m long leans against a vertical wall. If the foot of the ladder is 10 m away from the wall, what is the angle the ladder makes with the ground? (Use cos 60° = 0.5)

Accepted Answer

Step 1: Let the angle with the ground be θ. cos θ = adjacent/hypotenuse = 10/20 = 0.5. Step 2: cos θ = 0.5 → θ = 60°. Therefore, the ladder makes an angle of 60° with the ground.

Question 14

If sin θ = 3/5 and θ is an acute angle, find the value of tan θ.

Accepted Answer

Step 1: Given sin θ = 3/5. Using the Pythagorean identity sin²θ + cos²θ = 1, we get (3/5)² + cos²θ = 1. Step 2: 9/25 + cos²θ = 1, so cos²θ = 16/25, thus cos θ = 4/5 (positive since θ is acute). Step 3: tan θ = sin θ / cos θ = (3/5) ÷ (4/5) = 3/4.

Question 15

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 16

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 17

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 18

If cot θ = 7/24 and θ is in the third quadrant, find the value of (13 sin θ + 5 cos θ) / (13 sin θ - 5 cos θ).

Accepted Answer

Step 1: cot θ = 7/24, so cos θ / sin θ = 7/24. Step 2: Let cos θ = 7k, sin θ = 24k. Step 3: From sin²θ + cos²θ = 1: (24k)² + (7k)² = 625k² = 1 → k = ±1/25. Step 4: In Q3, both are negative: sin θ = -24/25, cos θ = -7/25. Step 5: Numerator = 13(-24/25) + 5(-7/25) = (-312 - 35)/25 = -347/25. Step 6: Denominator = 13(-24/25) - 5(-7/25) = (-312 + 35)/25 = -277/25. Step 7: Result = (-347/25) / (-277/25) = 347/277.

Question 19

If sin θ + cos θ = √2, find the value of sin³ θ + cos³ θ.

Accepted Answer

Step 1: Given sin θ + cos θ = √2. Step 2: Square both sides: (sin θ + cos θ)² = 2. Step 3: Expand: sin²θ + 2 sin θ cos θ + cos²θ = 2. Step 4: Simplify using sin²θ + cos²θ = 1: 1 + 2 sin θ cos θ = 2, so sin θ cos θ = 1/2. Step 5: Use the factorization sin³θ + cos³θ = (sin θ + cos θ)³ - 3 sin θ cos θ (sin θ + cos θ). Step 6: Substitute: sin³θ + cos³θ = (√2)³ - 3(1/2)(√2) = 2√2 - (3√2)/2 = (4√2 - 3√2)/2 = √2/2.

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