Question 1

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

Accepted Answer

Step 1: From tan θ = 5/12, we know opposite = 5 and adjacent = 12. Step 2: Use Pythagoras: hypotenuse² = 5² + 12² = 25 + 144 = 169, so hypotenuse = 13. Step 3: sec θ = hypotenuse/adjacent = 13/12.

Question 2

If cos θ = 7/25, find tan θ (where θ is acute).

Accepted Answer

Step 1: From cos θ = 7/25, adjacent = 7 and hypotenuse = 25. Step 2: Use Pythagoras: opposite² = 25² − 7² = 625 − 49 = 576, so opposite = 24. Step 3: tan θ = opposite/adjacent = 24/7.

Question 3

In a right triangle, sin θ = 4/5. What is the value of cot θ?

Accepted Answer

Step 1: From sin θ = 4/5, opposite = 4 and hypotenuse = 5. Step 2: Use Pythagoras: adjacent² = 5² − 4² = 25 − 16 = 9, so adjacent = 3. Step 3: cot θ = adjacent/opposite = 3/4.

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: sin θ is defined as opposite/hypotenuse. Step 2: sin θ = 8/17.

Question 5

A person standing on the ground observes the angle of elevation to the top of a tree to be 45°. If the person walks 10 metres closer to the tree, the angle of elevation becomes 60°. What is the height of the tree? (Use √3 = 1.732)

Accepted Answer

Step 1: Let height = h, initial distance = d. From first position: tan(45°) = h/d, so h = d. Step 2: From second position: tan(60°) = h/(d-10), so √3 = h/(d-10). Step 3: Substitute h = d: √3 = d/(d-10). Step 4: √3(d-10) = d. Step 5: √3·d - 10√3 = d. Step 6: d(√3 - 1) = 10√3. Step 7: d = 10√3/(√3-1) = 10√3(√3+1)/[(√3-1)(√3+1)] = 10(3+√3)/2 = 5(3+1.732) = 5 × 4.732 = 23.66 ≈ 24 metres. Step 8: h = d ≈ 24 metres.

Question 6

A boy standing 15 metres from a building observes the angle of elevation to the roof to be 60°. What is the height of the building? (Use √3 = 1.732)

Accepted Answer

Step 1: tan(60°) = height/distance. Step 2: tan(60°) = √3, so √3 = h/15. Step 3: h = 15√3 = 15 × 1.732 = 25.98 ≈ 26 metres.

Question 7

From a point on the ground 20 metres away from the base of a vertical pole, the angle of elevation to the top is 30°. What is the height of the pole? (Use √3 = 1.732)

Accepted Answer

Step 1: tan(30°) = height/distance. Step 2: tan(30°) = 1/√3, so 1/√3 = h/20. Step 3: h = 20/√3 = 20/1.732 = 11.55 ≈ 11.5 metres. Alternatively: h = 20/√3 × √3/√3 = 20√3/3 ≈ 11.5 metres.

Question 8

If tan θ = 1, what is the value of θ (in degrees)?

Accepted Answer

Step 1: We need to find the angle θ where tan θ = 1. Step 2: From the standard trigonometric values, tan 45° = 1. Step 3: Therefore, θ = 45°.

Question 9

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°. What is the height of the tower? (Use √3 = 1.732)

Accepted Answer

Step 1: In the right triangle formed, tan(60°) = height/distance. Step 2: tan(60°) = √3, so √3 = h/30. Step 3: h = 30√3 = 30 × 1.732 = 51.96 ≈ 52 metres.

Question 10

From the top of a 45-metre-high building, the angle of depression to a point on the ground is 30°. How far is the point from the base of the building?

Accepted Answer

Step 1: In the right triangle, tan(30°) = height/horizontal distance. Step 2: tan(30°) = 1/√3, so 1/√3 = 45/d. Step 3: d = 45√3 = 45 × 1.732 = 77.94 ≈ 78 metres.

Question 11

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

Accepted Answer

Step 1: Use the Pythagorean identity sin²θ + cos²θ = 1. Step 2: Substitute sin θ = 3/5: (3/5)² + cos²θ = 1. Step 3: 9/25 + cos²θ = 1, so cos²θ = 16/25. Step 4: Since θ is acute, cos θ = 4/5.

Question 12

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

Accepted Answer

Step 1: Observe that 62° + 28° = 90°, so these angles are complementary. Step 2: Apply the complementary angle property: 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 13

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: cos(θ) = sin(90° - θ). Step 3: Therefore, cos(48°) = sin(90° - 48°) = sin(42°). Step 4: Since cos(48°) = 0.67, we have sin(42°) = 0.67.

Question 14

If tan(3x) = cot(60°), where x is acute, find the value of x.

Accepted Answer

Step 1: Recognize that cot(60°) = 1/tan(60°) = 1/√3. Step 2: Apply the complementary angle property: cot(60°) = tan(90° - 60°) = tan(30°). Step 3: Therefore, tan(3x) = tan(30°). Step 4: Since x is acute, 3x = 30°. Step 5: Solve for x: x = 10°.

Question 15

Simplify: sin(25°)·sec(65°) + cos(25°)·csc(65°)

Accepted Answer

Step 1: Identify complementary angle pairs: 25° + 65° = 90°. Step 2: Apply complementary identities: sin(25°) = cos(65°) and cos(25°) = sin(65°). Step 3: Rewrite sec(65°) = 1/cos(65°) and csc(65°) = 1/sin(65°). Step 4: Substitute: sin(25°)·sec(65°) = cos(65°)·(1/cos(65°)) = 1. Step 5: Similarly, cos(25°)·csc(65°) = sin(65°)·(1/sin(65°)) = 1. Step 6: Total = 1 + 1 = 2.

Question 16

If 3 sin θ - 4 cos θ = 0, find the value of tan θ (where θ is acute).

Accepted Answer

Step 1: 3 sin θ - 4 cos θ = 0 implies 3 sin θ = 4 cos θ. Step 2: Divide both sides by cos θ: 3 tan θ = 4. Step 3: tan θ = 4/3.

Question 17

If cos θ = 7/25 and θ is acute, what is sin θ + tan θ?

Accepted Answer

Step 1: cos θ = 7/25 means adjacent = 7, hypotenuse = 25. Step 2: Find opposite using Pythagoras: opposite² + 7² = 25², so opposite² = 625 - 49 = 576, thus opposite = 24. Step 3: sin θ = 24/25. Step 4: tan θ = 24/7. Step 5: sin θ + tan θ = 24/25 + 24/7 = (24×7 + 24×25)/(25×7) = (168 + 600)/175 = 768/175.

Question 18

A man standing 40 metres away from the base of a tower observes the angle of elevation to the top of the tower as 60°. Find the height of the tower (in metres). [Use √3 = 1.732]

Accepted Answer

Step 1: Let height of tower = h metres. Distance from base = 40 metres. Angle of elevation = 60°. Step 2: Using tan(60°) = h/40, we get h = 40 × tan(60°) = 40 × √3. Step 3: h = 40 × 1.732 = 69.28 metres. Therefore, the height is approximately 69.28 metres.

Question 19

From a point on the ground, the angle of elevation to the top of a 30-metre tall building is 45°. How far is the point from the base of the building?

Accepted Answer

Step 1: Height of building = 30 metres. Angle of elevation = 45°. Let distance from base = d metres. Step 2: Using tan(45°) = height/distance, we get 1 = 30/d. Step 3: d = 30 metres. Therefore, the point is 30 metres away from the base.

Question 20

A ladder leans against a wall making an angle of 60° with the ground. If the ladder is 8 metres long, how high does it reach on the wall? [Use √3 = 1.732]

Accepted Answer

Step 1: Ladder length (hypotenuse) = 8 metres. Angle with ground = 60°. Height on wall = h metres. Step 2: Using sin(60°) = h/8, we get h = 8 × sin(60°) = 8 × (√3/2). Step 3: h = 8 × (1.732/2) = 8 × 0.866 = 6.928 metres. Therefore, the ladder reaches approximately 6.93 metres high.

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