Question 1

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

Accepted Answer

Step 1: Identify that 35° and 55° are complementary angles because 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 2

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

Accepted Answer

Step 1: Recall the complementary angle property of trigonometric functions: sin(A) = cos(90° - A), or equivalently, sin(A) = cos(B) when A + B = 90°.

Step 2: Given sin(35°) = cos(θ), we apply the complementary angle relationship.

Step 3: Using sin(35°) = cos(90° - 35°), we get sin(35°) = cos(55°).

Step 4: Comparing sin(35°) = cos(θ) with sin(35°) = cos(55°), we have θ = 55°.

Therefore, answer is 55°.

Question 3

A man standing 40 metres away from the base of a tower observes that the angle of elevation to the top of the tower is 30°. Find the height of the tower. (Use √3 ≈ 1.732)

Accepted Answer

Step 1: In the right triangle formed, we have the horizontal distance (adjacent) = 40 m and angle of elevation = 30°. Step 2: Using tan(30°) = height/distance, we get height = distance × tan(30°). Step 3: tan(30°) = 1/√3. Step 4: Height = 40 × (1/√3) = 40/√3. Step 5: Rationalizing: (40/√3) × (√3/√3) = 40√3/3 = (40 × 1.732)/3 = 69.28/3 ≈ 23.09 m. Step 6: Rounding to nearest practical value: approximately 23.1 m or 40/√3 m exactly.

Question 4

A man standing 30 metres away from the base of a tower observes the angle of elevation to the top of the tower to be 60°. Find the height of the tower. (Use √3 ≈ 1.732)

Accepted Answer

Step 1: Let the height of the tower be h metres. The man is standing 30 m away from the base (horizontal distance = 30 m). Step 2: Using the tangent ratio: tan(60°) = h/30. Step 3: Since tan(60°) = √3, we have √3 = h/30. Step 4: Therefore, h = 30√3 = 30 × 1.732 = 51.96 ≈ 52 metres.

Question 5

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

Accepted Answer

Step 1: Identify that 35° and 55° are complementary angles because 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 6

If cos θ = 5/13, what is the value of sin²θ?

Accepted Answer

Step 1: Use the identity sin²θ + cos²θ = 1, so sin²θ = 1 − cos²θ. Step 2: cos²θ = (5/13)² = 25/169. Step 3: sin²θ = 1 − 25/169 = (169 − 25)/169 = 144/169. Therefore, sin²θ = 144/169.

Question 7

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) when A + B = 90°. Step 2: Set up the equation: (5θ + 12°) + (3θ + 28°) = 90°. Step 3: Simplify: 8θ + 40° = 90°. Step 4: Solve for θ: 8θ = 50°, so θ = 6.25°. Therefore, the answer is 6.25°.

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° = 25/4 degrees.

Question 9

If sin θ = 5/13, find the value of (tan θ + cos θ).

Accepted Answer

Step 1: Given sin θ = 5/13. Using Pythagorean identity, cos θ = √(1 - 25/169) = √(144/169) = 12/13. Step 2: tan θ = sin θ / cos θ = (5/13) / (12/13) = 5/12. Step 3: tan θ + cos θ = 5/12 + 12/13 = (65 + 144) / 156 = 209/156. Therefore, the answer is 209/156.

Question 10

If sin(3A − 15°) = cos(A + 25°), where 3A − 15° and A + 25° are both acute angles, find the value of A.

Accepted Answer

Step 1: Use the complementary angle identity: sin θ = cos(90° − θ). So sin(3A − 15°) = cos(90° − (3A − 15°)) = cos(105° − 3A). Step 2: Set equal to cos(A + 25°): cos(105° − 3A) = cos(A + 25°). Step 3: Since cosine is one-to-one for acute angles: 105° − 3A = A + 25°. Step 4: 105° − 25° = A + 3A → 80° = 4A → A = 20°. Therefore, the answer is 20°.

Question 11

From the top of a lighthouse 60 m high, the angles of depression of two ships on the same side are 45° and 30°. What is the distance between the two ships?

Accepted Answer

Step 1: Let the lighthouse be AB = 60 m. For the nearer ship C, angle of depression = 45°. tan 45° = 60/BC → BC = 60 m. Step 2: For the farther ship D, angle of depression = 30°. tan 30° = 60/BD → BD = 60√3 m. Step 3: Distance between ships = BD − BC = 60√3 − 60 = 60(√3 − 1) m. Using √3 ≈ 1.732: 60 × 0.732 = 43.92 m. In exact form: 60(√3 − 1) m. Therefore, the answer is 60(√3 − 1) m.

Question 12

If sin θ + cos θ = 7/5, where θ is an acute angle, then find the value of tan θ + cot θ.

Accepted Answer

Step 1: Square both sides of sin θ + cos θ = 7/5 to get sin²θ + 2sin θ cos θ + cos²θ = 49/25. Step 2: Since sin²θ + cos²θ = 1, we have 1 + 2sin θ cos θ = 49/25, so 2sin θ cos θ = 24/25. Step 3: Therefore sin θ cos θ = 12/25. Step 4: tan θ + cot θ = (sin θ/cos θ) + (cos θ/sin θ) = (sin²θ + cos²θ)/(sin θ cos θ) = 1/(12/25) = 25/12.

Question 13

If sin(5x + 20°) = cos(3x + 40°), where both angles are acute, then find the value of tan(2x + 15°).

Accepted Answer

Step 1: Use the complementary angle property: sin θ = cos(90° - θ). Therefore, sin(5x + 20°) = cos(90° - 5x - 20°) = cos(70° - 5x). Step 2: Set this equal to the given expression: cos(70° - 5x) = cos(3x + 40°). Step 3: Since both angles are acute and cosine is one-to-one in [0°, 90°], we have: 70° - 5x = 3x + 40°. Step 4: Solve for x: 70° - 40° = 3x + 5x → 30° = 8x → x = 3.75°. Step 5: Find 2x + 15° = 2(3.75°) + 15° = 7.5° + 15° = 22.5°. Step 6: Calculate tan(22.5°) = √2 - 1 (this is a standard value). Alternatively, tan(22.5°) can be computed using the half-angle formula: tan(45°/2) = (1 - cos 45°)/sin 45° = (1 - 1/√2)/(1/√2) = √2 - 1.

Question 14

If sin θ + cos θ = 7/5, where θ is an acute angle, then find the value of tan θ + cot θ.

Accepted Answer

Step 1: Square both sides of sin θ + cos θ = 7/5 to get sin²θ + 2sinθcosθ + cos²θ = 49/25. Step 2: Since sin²θ + cos²θ = 1, we have 1 + 2sinθcosθ = 49/25, so 2sinθcosθ = 24/25, thus sinθcosθ = 12/25. Step 3: tan θ + cot θ = (sin θ/cos θ) + (cos θ/sin θ) = (sin²θ + cos²θ)/(sinθcosθ) = 1/(12/25) = 25/12.

Question 15

From a point on the ground 40 m away from the base of a vertical tower, the angle of elevation to the top is 60°. A man climbs to a point on the tower such that the angle of elevation from the same ground point becomes 45°. How many metres did the man climb?

Accepted Answer

Step 1: Let the height of the tower be H. From the ground point 40 m away, tan(60°) = H/40, so H = 40√3 m. Step 2: Let the man climb to height h. From the same ground point, tan(45°) = h/40, so h = 40 m. Step 3: Distance climbed = H − h = 40√3 − 40 = 40(√3 − 1) m ≈ 40(1.732 − 1) = 40(0.732) = 29.28 m. Rounding to nearest clean value: 40(√3 − 1) ≈ 29.3 m, or exactly 40(√3 − 1) m.

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