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Trigonometry

Mathematical Abilities — Paper-I · Official 2026 Coverage

Useful for: SSC CGL

Overview

Tier-II trigonometry adds identities, maximum–minimum values and two-angle height–distance problems.

Key points

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Maximum of a sin θ + b cos θ is √(a² + b²).
If tan θ = p/b, then sin θ = p/h and cos θ = b/h with h = √(p² + b²).
Complementary angles: sin(90° − θ) = cos θ.
Height problems: tan θ = height ÷ horizontal distance.

Explanation

Two observation points: write tan for each and subtract distances.
Minimum value of sin²θ + cos²θ forms is constant 1; watch for traps.
sec θ + tan θ = k → sec θ − tan θ = 1/k.
Keep √3 ≈ 1.732 for numeric answers.

Step by step

Identify the concept and the formula that links the given values.
Solve step by step, keeping units consistent.
Check the answer against the options and by a quick estimate.

Examples

tan θ = 3/4 → sin θ = 3/5, cos θ = 4/5 → sin θ + cos θ = 7/5.
Maximum of 3 sin θ + 4 cos θ = √(9 + 16) = 5.
A 50 m tower seen at 30° elevation: distance = 50/tan 30° = 50√3 ≈ 86.6 m.

Traps and advanced points

Trap: Using sin for height–distance instead of tan.
Trap: Forgetting the square root in the maximum formula.
Trap: Sign errors in identities.

Quick revision

√(a² + b²) maximum.
sec + tan = k → sec − tan = 1/k.
tan for heights.

Practice MCQs

From a point 78 m from the foot of a vertical pole, the angle of elevation of its top is 45°. Ignoring eye height, find the pole height.
1. 156 m
2. 78 m
3. 135.1 m
4. 39 m
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Answer: (b) 78 m. Use tanθ = height/base; tan45° = 1.
From a point 66 m from the foot of a vertical pole, the angle of elevation of its top is 45°. Ignoring eye height, find the pole height.
1. 132 m
2. 66 m
3. 33 m
4. 114.32 m
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Answer: (b) 66 m. Use tanθ = height/base; tan45° = 1.
Find 56 × sin 90° (use the standard trigonometric value).
1. 44.8
2. 67.2
3. 78.4
4. 56
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Answer: (d) 56. Use standard trigonometric values.
From a point 79 m from the foot of a vertical pole, the angle of elevation of its top is 30°. Ignoring eye height, find the pole height.
1. 45.61 m
2. 39.5 m
3. 158 m
4. 136.83 m
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Answer: (a) 45.61 m. Use tanθ = height/base; tan30° = 1/√3.
Find 3 × sin 30° (use the standard trigonometric value).
1. 0.5
2. 2.5
3. 3.5
4. 1.5
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Answer: (d) 1.5. Use standard trigonometric values.
Find 4 × sin 45° (use the standard trigonometric value).
1. 3.83
2. 4.83
3. 1.83
4. 2.83
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Answer: (d) 2.83. Use standard trigonometric values.
Find 5 × sin 60° (use the standard trigonometric value).
1. 6.33
2. 5.33
3. 4.33
4. 3.33
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Answer: (c) 4.33. Use standard trigonometric values.
Find 6 × sin 90° (use the standard trigonometric value).
1. 7.2
2. 4.8
3. 6
4. 8.4
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Answer: (c) 6. Use standard trigonometric values.
Find 7 × sin 0° (use the standard trigonometric value).
1. 1
2. 2
3. 0
4. -1
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Answer: (c) 0. Use standard trigonometric values.
Find 9 × sin 45° (use the standard trigonometric value).
1. 6.36
2. 5.09
3. 7.64
4. 8.91
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Answer: (a) 6.36. Use standard trigonometric values.
Find 10 × sin 60° (use the standard trigonometric value).
1. 10.39
2. 12.12
3. 8.66
4. 6.93
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Answer: (c) 8.66. Use standard trigonometric values.
Find 67 × sin 0° (use the standard trigonometric value).
1. 0
2. 1
3. 2
4. -1
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Answer: (a) 0. Use standard trigonometric values.
Find 12 × sin 0° (use the standard trigonometric value).
1. 1
2. -1
3. 2
4. 0
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Answer: (d) 0. Use standard trigonometric values.
Find 13 × sin 30° (use the standard trigonometric value).
1. 5.2
2. 7.8
3. 6.5
4. 9.1
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Answer: (c) 6.5. Use standard trigonometric values.
Find 15 × sin 60° (use the standard trigonometric value).
1. 15.59
2. 10.39
3. 18.19
4. 12.99
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Answer: (d) 12.99. Use standard trigonometric values.