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SSC CPO Complementary Angles

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This page covers SSC CPO Complementary Angles with complete concept notes, 10 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Complementary Angles— Rules & Concept

Core ConceptRead this first — the foundation of the topic
SSC CGL typically asks three types of questions

direct formula application, simplification problems, and value finding. Direct questions give you an angle and ask for complementary ratio values. Simplification problems mix multiple complementary ratios in expressions. Value finding questions provide ratio values and ask for complementary angle ratios.

Here's a powerful shortcut: whenever you see (90° - θ) in any trigonometric expression, immediately swap the ratio with its complementary partner. Sin becomes cos, tan becomes cot, sec becomes cosec. This works both ways. If you see sin 60°, think cos 30°.

If you see tan 25°, think cot 65°

Let's solve a typical example
1

Identify complementary angles. Since 30° + 60° = 90°, these are complementary.

2

Apply complementary relationships. cos 60° = cos(90° - 30°) = sin 30°, sin 60° = sin(90° - 30°) = cos 30°.

3

Substitute values. sin 30° × sin 30° + cos 30° × cos 30° = (sin 30°)² + (cos 30°)².

4

Use standard values. sin 30° = 1/2, cos 30° = √3/2.

5

Calculate final answer. (1/2)² + (√3/2)² = 1/4 + 3/4 = 1. Another shortcut involves recognizing patterns. Expression like sin²θ + cos²θ always equals 1. When you see complementary angles multiplied or added in specific patterns, look for these standard identities.

Exam TrapsCommon mistakes students make — avoid these

include forgetting that complementary means sum equals 90°, not 180°. Students often confuse complementary with supplementary angles. Another error is applying relationships incorrectly - remember sin(90° - θ) = cos θ, not sin θ = cos(90° - θ).

Practice identifying complementary pairs quickly: 30°-60°, 45°-45°, 25°-65°, 37°-53°. Most SSC questions use these standard combinations.

Key Points to Remember

  • Complementary angles sum to exactly 90 degrees
  • sin(90° - θ) = cos θ and cos(90° - θ) = sin θ
  • tan(90° - θ) = cot θ and cot(90° - θ) = tan θ
  • sec(90° - θ) = cosec θ and cosec(90° - θ) = sec θ
  • Common complementary pairs: 30°-60°, 45°-45°, 37°-53°
  • Quick trick: swap ratios when you see (90° - θ)
  • sin²θ + cos²θ = 1 for any complementary relationship
  • Complementary means 90°, supplementary means 180°

Exam-Specific Tips

  • sin 30° = 1/2 and cos 60° = 1/2
  • tan 45° = cot 45° = 1
  • sin 37° = cos 53° = 3/5
  • cos 37° = sin 53° = 4/5
  • sin²θ + cos²θ = 1 for all values of θ
  • sec θ × cos θ = 1 for complementary calculations
  • Standard complementary pairs sum to 90°: 30°-60°, 45°-45°, 37°-53°
  • cosec(90° - θ) = sec θ identity appears in 15% of trigonometry questions
Practice MCQs

Complementary Angles — Practice Questions

10graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

If sin(35°) = k, then cos(55°) = ?

Practice 2medium

Evaluate: [sin²(23°) + sin²(67°)] / [cos²(23°) + cos²(67°)]

Practice 3medium

If tan(5α) = cot(4α) and 0° < α < 18°, find α.

Practice 4medium

Given tan(62°) = m, express cot(28°) in terms of m.

Practice 5medium

If sin(3x) = cos(2x), where 0° < x < 90°, find the value of x.

Practice 6medium

Simplify: sin(40°)·sec(50°) + cos(40°)·csc(50°)

Practice 7medium

If sin(35°) = k, then what is the value of cos(55°)?

Practice 8hard

If sec(θ) = cosec(θ + 20°) where θ is acute, find the value of sin(2θ + 10°).

Practice 9hard

Given that tan(α) · tan(90° - α) · tan(2α) = √3, where 0° < α < 45°, find α.

Practice 10hard

If sin²(β) + sin²(90° - β) = k, find the value of k.

60-Second Revision — Complementary Angles

  • Remember: Complementary angles add to 90°, not 180°
  • Formula: sin(90° - θ) = cos θ, cos(90° - θ) = sin θ
  • Shortcut: When you see (90° - θ), swap the trigonometric ratio
  • Key pairs: 30°-60°, 45°-45°, 37°-53° appear most frequently
  • Identity: sin²θ + cos²θ = 1 works for all complementary problems
  • Trap: Don't confuse complementary (90°) with supplementary (180°)
  • Quick check: Verify angle sum equals 90° before applying formulas
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