Question 1

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 2

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 3

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 4

If sin 30° = 1/2, what is the value of cos 60°?

Accepted Answer

Step 1: Recall the complementary angle property: sin θ = cos(90° - θ). Step 2: sin 30° = cos(90° - 30°) = cos 60°. Step 3: Therefore, cos 60° = 1/2. The answer is 1/2.

Question 5

In a right-angled triangle, if sin A = 4/5, what is the value of tan A?

Accepted Answer

Step 1: Use sin A = 4/5 and the Pythagorean identity sin²A + cos²A = 1. Step 2: (4/5)² + cos²A = 1 → 16/25 + cos²A = 1. Step 3: cos²A = 9/25 → cos A = 3/5. Step 4: tan A = sin A / cos A = (4/5) / (3/5) = 4/3. Therefore, the answer is 4/3.

Question 6

If tan 45° = 1, what is the value of cot 45°?

Accepted Answer

Step 1: Recall that cot θ = 1/tan θ. Step 2: cot 45° = 1/tan 45° = 1/1 = 1. Therefore, the answer is 1.

Question 7

If cot θ = 7/24 and θ is in the first quadrant, find sin θ.

Accepted Answer

Step 1: cot θ = 7/24 means adjacent/opposite = 7/24 in a right triangle. Step 2: Let opposite = 24 and adjacent = 7. Step 3: Find hypotenuse using Pythagoras: h² = 7² + 24² = 49 + 576 = 625, so h = 25. Step 4: sin θ = opposite/hypotenuse = 24/25. Therefore, answer is 24/25.

Question 8

In a right-angled triangle, the hypotenuse is 13 cm and one side is 5 cm. If θ is the angle opposite the side of length 5 cm, what is sin θ?

Accepted Answer

Step 1: Use the Pythagorean theorem to find the third side: a² + 5² = 13². Step 2: a² + 25 = 169. Step 3: a² = 144, so a = 12 cm. Step 4: sin θ = opposite/hypotenuse = 5/13. Therefore, the answer is 5/13.

Question 9

In a right-angled triangle, if one acute angle is 30°, what is the value of sin 30°?

Accepted Answer

Step 1: Recall the standard value of sin 30° from the special angles table. Step 2: sin 30° = 1/2.

Question 10

If cos θ = 8/17 and θ is acute, what is the value of tan θ?

Accepted Answer

Step 1: cos θ = 8/17 means adjacent/hypotenuse = 8/17. Step 2: Let adjacent = 8 and hypotenuse = 17. Step 3: Find opposite using Pythagoras: opposite² = 17² - 8² = 289 - 64 = 225, so opposite = 15. Step 4: tan θ = opposite/adjacent = 15/8. Therefore, answer is 15/8.

Question 11

If sec θ - tan θ = 2/5, 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 = 2/5: (2/5)(sec θ + tan θ) = 1. Step 3: sec θ + tan θ = 1 ÷ (2/5) = 5/2.

Question 12

In a right-angled triangle, if tan α = 7/24, find the value of (sin α + cos α) / (sin α - cos α).

Accepted Answer

Step 1: tan α = 7/24 means opposite = 7, adjacent = 24. Using Pythagoras: hypotenuse = √(7² + 24²) = √(49 + 576) = √625 = 25. Step 2: sin α = 7/25, cos α = 24/25. Step 3: (sin α + cos α) / (sin α - cos α) = (7/25 + 24/25) / (7/25 - 24/25) = (31/25) / (-17/25) = 31 / (-17) = -31/17.

Question 13

If tan θ = 5/12 and θ is in the first quadrant, find the value of (13 sin θ - 5 cos θ).

Accepted Answer

Step 1: tan θ = 5/12 means opposite = 5, adjacent = 12. Using Pythagoras: hypotenuse = √(25 + 144) = √169 = 13. Step 2: sin θ = 5/13 and cos θ = 12/13. Step 3: 13 sin θ - 5 cos θ = 13(5/13) - 5(12/13) = 5 - 60/13 = (65 - 60)/13 = 5/13.

Question 14

If sin θ = cos θ, where 0° < θ < 90°, find the value of (sin³θ + cos³θ + 2 sin θ cos θ).

Accepted Answer

Step 1: sin θ = cos θ implies tan θ = 1, so θ = 45°. Step 2: sin 45° = cos 45° = 1/√2. Step 3: sin³45° = (1/√2)³ = 1/(2√2) and cos³45° = 1/(2√2). Step 4: sin 45° cos 45° = (1/√2)(1/√2) = 1/2. Step 5: sin³θ + cos³θ + 2 sin θ cos θ = 1/(2√2) + 1/(2√2) + 2(1/2) = 2/(2√2) + 1 = 1/√2 + 1 = (1 + √2)/√2 = (√2 + 2)/2.

Question 15

If tan A = 1 and tan B = 1/√3, where A and B are acute angles, find the value of (A + B) in degrees.

Accepted Answer

Step 1: tan A = 1 implies A = 45°. Step 2: tan B = 1/√3 implies B = 30° (since tan 30° = 1/√3). Step 3: A + B = 45° + 30° = 75°.

Question 16

In a right triangle, if cot θ = 8/15, find the value of cosec θ.

Accepted Answer

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

Question 17

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

Accepted Answer

Step 1: Square both sides: (sin θ + cos θ)² = (√2)². Step 2: Expand: sin²θ + 2 sin θ cos θ + cos²θ = 2. Step 3: Use sin²θ + cos²θ = 1: 1 + 2 sin θ cos θ = 2. Step 4: Solve: 2 sin θ cos θ = 1, so sin θ cos θ = 1/2.

Question 18

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 19

In a right-angled triangle, if tan A = 5/12, find the value of sec A.

Accepted Answer

Step 1: tan A = 5/12 means opposite/adjacent = 5/12. Step 2: Using Pythagorean theorem, hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13. Step 3: sec A = hypotenuse/adjacent = 13/12.

Question 20

A ladder leans against a wall making an angle of 60° with the ground. If the ladder is 10 metres long, how far is the base of the ladder from the wall?

Accepted Answer

Step 1: The ladder, wall, and ground form a right triangle. The ladder is the hypotenuse (10 m), and the angle with the ground is 60°. Step 2: The distance from the base to the wall is the adjacent side to the 60° angle. Step 3: cos 60° = adjacent / hypotenuse = distance / 10. Step 4: cos 60° = 1/2, so distance = 10 × (1/2) = 5 metres.

Question 21

If cos θ − sin θ = √2 sin θ, where 0° < θ < 90°, find the value of (cos θ + sin θ)/(cos θ − sin θ).

Accepted Answer

Step 1: Given cos θ − sin θ = √2 sin θ. Rearrange: cos θ = sin θ + √2 sin θ = (1 + √2) sin θ. Step 2: Use sin² θ + cos² θ = 1: sin² θ + [(1 + √2) sin θ]² = 1 → sin² θ + (1 + 2√2 + 2) sin² θ = 1 → sin² θ (1 + 3 + 2√2) = 1 → sin² θ (4 + 2√2) = 1 → sin² θ = 1/(4 + 2√2). Step 3: Rationalize: sin² θ = 1/(4 + 2√2) × (4 − 2√2)/(4 − 2√2) = (4 − 2√2)/(16 − 8) = (4 − 2√2)/8 = (2 − √2)/4. Step 4: sin θ = √[(2 − √2)/4] = √(2 − √2)/2. Step 5: cos θ = (1 + √2) sin θ = (1 + √2) × √(2 − √2)/2. Step 6: cos θ + sin θ = sin θ [(1 + √2) + 1] = sin θ (2 + √2). Step 7: cos θ − sin θ = √2 sin θ (given). Step 8: (cos θ + sin θ)/(cos θ − sin θ) = [sin θ (2 + √2)] / [√2 sin θ] = (2 + √2)/√2 = (2 + √2)/√2 × √2/√2 = (2√2 + 2)/2 = √2 + 1. Therefore, answer is √2 + 1.

Question 22

If 3 sin θ + 4 cos θ = 5, find the value of 3 cos θ − 4 sin θ.

Accepted Answer

Step 1: Let A = 3 sin θ + 4 cos θ = 5 and B = 3 cos θ − 4 sin θ (unknown). Step 2: Square A: (3 sin θ + 4 cos θ)² = 25 → 9 sin² θ + 24 sin θ cos θ + 16 cos² θ = 25. Step 3: Square B: (3 cos θ − 4 sin θ)² = 9 cos² θ − 24 sin θ cos θ + 16 sin² θ. Step 4: Add A² and B²: 9(sin² θ + cos² θ) + 16(sin² θ + cos² θ) + 24 sin θ cos θ − 24 sin θ cos θ = 25 + B². Step 5: Simplify: 9 + 16 = 25 + B² → 25 = 25 + B² → B² = 0 → B = 0.

Question 23

If 3 sin θ - 4 cos θ = 5, find the value of 3 cos θ + 4 sin θ.

Accepted Answer

Step 1: Given 3 sin θ - 4 cos θ = 5. Step 2: Let 3 cos θ + 4 sin θ = k (to find). Step 3: Square the given equation: (3 sin θ - 4 cos θ)² = 25 → 9 sin² θ - 24 sin θ cos θ + 16 cos² θ = 25. Step 4: Square the expression to find: (3 cos θ + 4 sin θ)² = k² → 9 cos² θ + 24 sin θ cos θ + 16 sin² θ = k². Step 5: Add both squared equations: 9(sin² θ + cos² θ) + 16(sin² θ + cos² θ) = 25 + k² → 9 + 16 = 25 + k² → 25 = 25 + k² → k² = 0. Step 6: Therefore, k = 0.

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