1. From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. Calculate the distance of point P from the foot of the building.
Answer: Let’s solve this step-by-step: 1) Let the distance of point P from the foot of the building be x meters. 2) We know the height of the building is 10 m. 3) The angle of elevation is 30°. 4) In this right-angled triangle, tan 30° = opposite / adjacent = 10 / x 5) We know that tan 30° = 1/√3 6) So, 1/√3 = 10 / x 7) x = 10√3 ≈ 17.32 m Therefore, the distance of point P from the foot of the building is approximately 17.32 meters.
2. If sin θ = 4/5, find the value of cos θ and tan θ.
Answer: Let’s solve this step-by-step: 1) We know that sin² θ + cos² θ = 1 2) Given sin θ = 4/5, so sin² θ = 16/25 3) Therefore, cos² θ = 1 – 16/25 = 9/25 4) cos θ = 3/5 (we take the positive value as cos θ is positive in the first quadrant) 5) Now, tan θ = sin θ / cos θ = (4/5) / (3/5) = 4/3 Therefore, cos θ = 3/5 and tan θ = 4/3
3. A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.
Answer: Let’s solve this step-by-step: 1) Let the height of the tower be h meters. 2) We know the distance from the observer to the foot of the tower is 15 m. 3) The angle of elevation is 60°. 4) In this right-angled triangle, tan 60° = height / base 5) We know that tan 60° = √3 6) So, √3 = h / 15 7) h = 15√3 ≈ 25.98 m Therefore, the height of the tower is approximately 25.98 meters.
4. If sin A = 3/5, calculate the values of cos A and tan A.
Answer: Let’s solve this step-by-step: 1) We know that sin² A + cos² A = 1 2) Given sin A = 3/5, so sin² A = 9/25 3) Therefore, cos² A = 1 – 9/25 = 16/25 4) cos A = 4/5 (we take the positive value as cos A is positive in the first quadrant) 5) Now, tan A = sin A / cos A = (3/5) / (4/5) = 3/4 Therefore, cos A = 4/5 and tan A = 3/4
5. In a right triangle ABC, right-angled at B, if tan A = 1/√3, find the values of sin A and cos A.
Answer: Let’s approach this step-by-step: 1) We know tan A = 1/√3 2) In a right triangle, tan A = opposite / adjacent = sin A / cos A 3) Let’s assume sin A = x and cos A = y 4) Then, x/y = 1/√3 5) x = y/√3 6) We know sin² A + cos² A = 1 7) Substituting, (y/√3)² + y² = 1 8) (y²/3) + y² = 1 9) (1/3 + 1)y² = 1 10) (4/3)y² = 1 11) y² = 3/4 12) y = √(3/4) = √3/2 13) Therefore, cos A = √3/2 14) And sin A = 1/2 (because sin A = y/√3 = (√3/2)/√3 = 1/2) Thus, sin A = 1/2 and cos A = √3/2
6. A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.
Answer: Let’s solve this step-by-step: 1) Let the length of the ladder be x meters. 2) The distance of the foot of the ladder from the wall is 2.5 m. 3) The height of the window is 6 m. 4) This forms a right-angled triangle where the ladder is the hypotenuse. 5) We can use the Pythagorean theorem: a² + b² = c² 6) Here, 2.5² + 6² = x² 7) 6.25 + 36 = x² 8) 42.25 = x² 9) x = √42.25 ≈ 6.5 m Therefore, the length of the ladder is approximately 6.5 meters.
7. If cos A = 5/13, find the values of sin A and tan A.
Answer: Let’s solve this step-by-step: 1) We’re given that cos A = 5/13 2) In a right triangle, sin² A + cos² A = 1 3) Substituting the known value: sin² A + (5/13)² = 1 4) sin² A + 25/169 = 1 5) sin² A = 1 – 25/169 = 144/169 6) sin A = 12/13 (we take the positive value as sin A is positive in the first quadrant) 7) Now, tan A = sin A / cos A = (12/13) / (5/13) = 12/5 Therefore, sin A = 12/13 and tan A = 12/5