Trigonometry for Aptitude Tests

Trigonometry appears in placement tests mainly through height-and-distance problems and equation solving. Master the core identities and a few standard angles.

Standard Angles Table

Memory trick: For sin, read values top-to-bottom: 0, 1/2, 1/√2, √3/2, 1. For cos, read the same values bottom-to-top.

Fundamental Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Key Formulas

Heights and Distances

These problems involve a single right triangle with an angle of elevation or depression.

Terms:

• Angle of elevation: Looking up from horizontal.
• Angle of depression: Looking down from horizontal.

Example: A tower stands on level ground. From a point 100m from its base, the angle of elevation to the top is 30°. Find the tower’s height.

`` tan 30° = height / 100 height = 100 × (1/√3) = 100√3 / 3 ≈ 57.74 m

**Example 2:** From the top of a 50m building, the angle of depression of a car is 60°. Find the car's distance from the building.

``
tan 60° = 50 / distance
distance = 50 / √3 = 50√3 / 3 ≈ 28.87 m

Shortcuts

• Complementary angles: sin θ = cos (90° - θ) and tan θ = cot (90° - θ).
• Negative angles: sin(-θ) = -sin θ, cos(-θ) = cos θ.
• If sin θ + cos θ = 1, then sin θ cos θ = 0, so θ = 0° or 90°.

Practice Questions

Q1: A ladder 10m long leans against a wall making an angle of 60° with the ground. How high does it reach?

sin 60° = height / 10
height = 10 × √3/2 = 5√3 ≈ 8.66 m

Q2: If sin θ = 3/5, find cos θ.

cos θ = √(1 - sin²θ) = √(1 - 9/25) = √(16/25) = 4/5

Q3: Evaluate: sin² 45° + cos² 45°.

= (1/2) + (1/2) = 1  (by the identity sin²θ + cos²θ = 1)

Cross-references

• Calculus basics
• Number systems
• Logical reasoning
• Percentages