Question 1

A ladder leans against a wall such that it makes an angle of 60° with the ground. If the ladder is 10 metres long, find the height at which the top of the ladder touches the wall (in metres).

Accepted Answer

Step 1: The ladder forms the hypotenuse of a right triangle. The angle with the ground is 60°. Step 2: The height on the wall is the side opposite to the 60° angle. Using sin 60° = opposite/hypotenuse: sin 60° = height/10. Step 3: sin 60° = √3/2, so height = 10 × (√3/2) = 5√3 metres. Step 4: 5√3 ≈ 5 × 1.732 = 8.66 metres. For a clean answer, let's use the exact form: 5√3 metres.

Question 2

If sin θ = 3/5 and θ lies in the second quadrant, find the value of (5 tan θ + 4 sec θ).

Accepted Answer

Step 1: Since sin θ = 3/5 and θ is in Q2, use sin²θ + cos²θ = 1 to find cos θ. Step 2: (3/5)² + cos²θ = 1 → cos²θ = 16/25 → cos θ = -4/5 (negative in Q2). Step 3: tan θ = sin θ / cos θ = (3/5) / (-4/5) = -3/4. Step 4: sec θ = 1/cos θ = -5/4. Step 5: 5 tan θ + 4 sec θ = 5(-3/4) + 4(-5/4) = -15/4 - 20/4 = -35/4 = -8.75.

Question 3

Given that tan α = 1/2 and tan β = 1/3, where α and β are acute angles, find sin(α + β).

Accepted Answer

Step 1: Use tan(α + β) = (tan α + tan β) / (1 - tan α tan β) = (1/2 + 1/3) / (1 - 1/6) = (5/6) / (5/6) = 1. Step 2: Since tan(α + β) = 1 and both angles are acute, α + β = 45°. Step 3: sin(45°) = 1/√2 = √2/2.

Question 4

A ladder leans against a wall such that it makes an angle of 60° with the ground. If the ladder is 10 metres long, find the height at which the ladder touches the wall (in metres).

Accepted Answer

Step 1: The ladder forms the hypotenuse of a right triangle. The angle with the ground is 60°. Step 2: The height on the wall is the side opposite to the 60° angle. Step 3: sin(60°) = height / hypotenuse → height = 10 × sin(60°) = 10 × (√3/2) = 5√3 metres ≈ 8.66 metres.

Question 5

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

Accepted Answer

Step 1: Square both sides of sin θ + cos θ = √2. Step 2: sin²θ + 2 sin θ cos θ + cos²θ = 2. Step 3: Using sin²θ + cos²θ = 1, we get 1 + 2 sin θ cos θ = 2. Step 4: 2 sin θ cos θ = 1 → sin θ cos θ = 1/2.

Question 6

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

Accepted Answer

Step 1: Recognize that 3 sin θ + 4 cos θ = 5 is a linear combination. The maximum value of a sin θ + b cos θ is √(a² + b²). Step 2: √(3² + 4²) = √(9 + 16) = √25 = 5. Step 3: Since the equation equals its maximum, the expression achieves its peak. This occurs when sin θ / 3 = cos θ / 4 (proportionality condition). Step 4: tan θ = sin θ / cos θ = 3/4.

SBI PO Basic Trig Ratios

Basic Trig Ratios— Rules & Concept

Key Points to Remember

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Basic Trig Ratios — Practice Questions

60-Second Revision — Basic Trig Ratios