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Trigonometric Ratios and Identities Questions with Solutions

Unlike Height and Distance, which applies trigonometry to real-world triangles, this topic works with the ratios and identities directly — given one ratio, find another, or simplify an expression using a fixed identity. Here’s the concept, four fully worked examples, and a few to try yourself.

The concept, quickly

  • The core identity: sin²θ + cos²θ = 1, for any angle θ. Given one of sin θ or cos θ, this gives the other directly.
  • Two derived identities, both built from the core one: 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.
  • Standard angle values worth knowing by heart: sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = 1/√2; tan 45° = 1; tan 30° = 1/√3; tan 60° = √3.
  • For an acute angle, all ratios are positive, so taking the square root in an identity only ever needs the positive root — this simplifies working with sin²θ + cos²θ = 1 in practice.

Worked examples

1. If sin θ = 3/5 and θ is acute, find cos θ.

Solution

  1. Using sin²θ + cos²θ = 1: cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25.
  2. cos θ = √(16/25) = 4/5 (positive, since θ is acute).

Answer: 4/5

2. Evaluate: 2 sin²45° + 3 cos²45°

Solution

  1. sin 45° = cos 45° = 1/√2, so sin²45° = cos²45° = 1/2.
  2. 2(1/2) + 3(1/2) = 1 + 1.5 = 2.5.

Answer: 2.5

3. If tan θ = 3/4 and θ is acute, find sec θ.

Solution

  1. Using 1 + tan²θ = sec²θ: sec²θ = 1 + (3/4)² = 1 + 9/16 = 25/16.
  2. sec θ = √(25/16) = 5/4.

Answer: 5/4

4. Evaluate: (sin 30° + cos 60°) / (1 + tan 45°)

Solution

  1. sin 30° = 1/2 and cos 60° = 1/2, so the numerator = 1/2 + 1/2 = 1.
  2. tan 45° = 1, so the denominator = 1 + 1 = 2.
  3. Result = 1/2.

Answer: 1/2

Try these yourself

Apply the identity or the standard angle values, then check against the answer.

1. If sin θ = 5/13 and θ is acute, find cos θ.

Answer: 12/13

2. If tan θ = 5/12 and θ is acute, find sec θ.

Answer: 13/12

3. Evaluate: 3 sin²60° + 2 cos²30°

Answer: 3.75

Where this comes up

Trigonometric Ratios and Identities is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk — most often in SSC CGL, and underlies every Height and Distance question.

Practice more Trigonometric Ratios and Identities questions

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