Question 1

Simplify: (1 - sin²θ) / cos θ

Accepted Answer

Step 1: Use the identity 1 - sin²θ = cos²θ. Step 2: Substitute: cos²θ / cos θ. Step 3: Simplify by cancelling one cos θ: cos θ.

Question 2

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

Accepted Answer

Step 1: Use the trigonometric ratio tan(θ) = opposite/adjacent. Here, tan(30°) = height/40. Step 2: We know tan(30°) = 1/√3. Step 3: Therefore, height = 40 × (1/√3) = 40/√3. Step 4: Rationalize: (40/√3) × (√3/√3) = 40√3/3 ≈ (40 × 1.732)/3 ≈ 69.28/3 ≈ 23.09 metres. Step 5: Rounding to nearest practical value gives approximately 23.1 metres or 40/√3 metres exactly.

Question 3

An observer on the ground sees the top of a tree at an angle of elevation of 30°. If the observer moves 20 metres closer to the tree, the angle of elevation becomes 60°. Find the height of the tree.

Accepted Answer

Step 1: Let initial distance = x, height = h. Step 2: tan(30°) = h/x → h = x/√3. Step 3: tan(60°) = h/(x−20) → h = √3(x−20). Step 4: x/√3 = √3(x−20) → x = 3(x−20) → x = 3x − 60 → x = 30. Step 5: h = 30/√3 = 10√3 metres.

Question 4

In a right triangle, if the opposite side is 8 cm and the hypotenuse is 17 cm, find sin θ.

Accepted Answer

Step 1: Recall that sin θ = opposite/hypotenuse. Step 2: sin θ = 8/17. Step 3: This fraction is already in simplest form. Therefore, the answer is 8/17.

Question 5

If tan θ = 5/12, find sin θ (where θ is acute).

Accepted Answer

Step 1: Recall that tan θ = opposite/adjacent = 5/12. Step 2: Use the Pythagorean theorem to find the hypotenuse: h² = 5² + 12² = 25 + 144 = 169, so h = 13. Step 3: sin θ = opposite/hypotenuse = 5/13.

Question 6

A man standing on the ground observes the angle of elevation to the top of a cliff as 45°. He then walks 50 metres away from the cliff, and the angle of elevation becomes 30°. Find the height of the cliff.

Accepted Answer

Step 1: Let the initial distance from the cliff be x metres and height be h metres. Step 2: From the first position: tan(45°) = h/x → 1 = h/x → h = x. Step 3: From the second position (50 metres further): tan(30°) = h/(x+50) → 1/√3 = h/(x+50) → h = (x+50)/√3. Step 4: Equate both expressions: x = (x+50)/√3 → x√3 = x + 50 → x(√3−1) = 50 → x = 50/(√3−1). Step 5: Rationalize: x = 50(√3+1)/[(√3−1)(√3+1)] = 50(√3+1)/(3−1) = 50(√3+1)/2 = 25(√3+1) ≈ 25(2.732) ≈ 68.3 metres. Step 6: h = x = 25(√3+1) metres ≈ 68.3 metres.

Question 7

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

Accepted Answer

Step 1: Use the identity 1 + tan²θ = sec²θ. Step 2: Substitute tan θ = 12/5: 1 + (12/5)² = sec²θ. Step 3: 1 + 144/25 = sec²θ. Step 4: (25 + 144)/25 = sec²θ. Step 5: 169/25 = sec²θ. Step 6: sec θ = 13/5 (positive in first quadrant). Therefore, the answer is 13/5.

Question 8

A ladder leans against a wall such that its foot is 9 metres from the wall. The angle of elevation from the foot of the ladder to the top of the wall is 60°. Find the height of the wall.

Accepted Answer

Step 1: The ladder's foot is 9 metres from the wall (horizontal distance). Step 2: The angle of elevation is 60°. Step 3: Use tan(60°) = height/distance. Step 4: tan(60°) = √3. Step 5: √3 = height/9. Step 6: height = 9√3 metres ≈ 9 × 1.732 ≈ 15.59 metres.

Question 9

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

Accepted Answer

Step 1: Recognize that 36° and 54° are complementary angles (36° + 54° = 90°). Step 2: Apply the complementary angle property: sec(θ) = cosec(90° − θ). Step 3: Therefore, sec(36°) = cosec(90° − 36°) = cosec(54°). Step 4: Since sec(36°) = m, we have cosec(54°) = m.

Question 10

sin(15°) × cosec(75°) + cos(15°) × sec(75°) = ?

Accepted Answer

Step 1: Recognize that 15° and 75° are complementary angles (15° + 75° = 90°). Step 2: Apply complementary angle properties: cosec(75°) = sec(15°) and sec(75°) = cosec(15°). Step 3: Substitute: sin(15°) × sec(15°) + cos(15°) × cosec(15°). Step 4: Simplify: sin(15°)/cos(15°) + cos(15°)/sin(15°) = tan(15°) + cot(15°). Step 5: Alternatively, use: sin(15°) × sec(15°) = sin(15°)/cos(15°) = tan(15°) and cos(15°) × cosec(15°) = cos(15°)/sin(15°) = cot(15°). Step 6: tan(15°) + cot(15°) = sin²(15°) + cos²(15°) / [sin(15°)cos(15°)] = 1 / [sin(15°)cos(15°)] = 2/sin(30°) = 2/(1/2) = 4.

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 property: 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

If cot θ = 5/12, find the value of csc θ (where θ is acute).

Accepted Answer

Step 1: Use the identity 1 + cot²θ = csc²θ. Step 2: Substitute cot θ = 5/12: 1 + (5/12)² = csc²θ. Step 3: 1 + 25/144 = csc²θ. Step 4: (144 + 25)/144 = csc²θ. Step 5: 169/144 = csc²θ. Step 6: csc θ = 13/12 (positive since θ is acute). Therefore, the answer is 13/12.

Question 13

tan(28°) × tan(62°) = ?

Accepted Answer

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

Question 14

Simplify: sec²θ - tan²θ

Accepted Answer

Step 1: Recall the Pythagorean identity sec²θ - tan²θ = 1. Step 2: This is a standard trigonometric identity that directly simplifies to 1.

Question 15

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

Accepted Answer

Step 1: Recognize that 42° and 48° are complementary angles (42° + 48° = 90°). Step 2: Apply the complementary angle property: 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 16

If tan θ = 1, what is the value of θ in degrees (where 0° < θ < 90°)?

Accepted Answer

Step 1: Recall that tan θ = opposite/adjacent. Step 2: tan θ = 1 means opposite = adjacent. Step 3: This occurs in a 45-45-90 triangle. Step 4: Therefore, θ = 45°. Therefore, the answer is 45°.

Question 17

cot(67°) = tan(x°). Find the value of x.

Accepted Answer

Step 1: Recall that cot(θ) = tan(90° − θ). Step 2: Apply this property: cot(67°) = tan(90° − 67°) = tan(23°). Step 3: Therefore, tan(x°) = tan(23°), which means x = 23.

Question 18

From a point on the ground 60 metres away from a building, the angle of elevation to the top is 45°. What is the height of the building?

Accepted Answer

Step 1: Use tan(45°) = height/distance. Step 2: tan(45°) = 1. Step 3: Therefore, 1 = height/60. Step 4: height = 60 metres.

Question 19

If cos(18°) = p, what is sin²(72°) + cos²(18°) equal to?

Accepted Answer

Step 1: Recognize that 18° + 72° = 90°, so these angles are complementary. Step 2: Apply the complementary angle identity: sin(72°) = sin(90° - 18°) = cos(18°) = p. Step 3: Rewrite the expression: sin²(72°) + cos²(18°) = p² + p². Step 4: Simplify: p² + p² = 2p².

Question 20

If cot(47°) = m, express tan(43°) in terms of m.

Accepted Answer

Step 1: Observe that 47° + 43° = 90°, so these angles are complementary. Step 2: Apply the complementary angle identity: tan(θ) = cot(90° - θ). Step 3: Therefore, tan(43°) = cot(90° - 43°) = cot(47°). Step 4: Since cot(47°) = m, we have tan(43°) = m.

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