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Trigonometry

Mathematics · Algebra, Geometry & Data

Useful for: Railway RRB, OSSC RHT — Regular High School Teacher, UPSC NDA/NA, UPSC CDS, OSEPA Primary Teacher / JT Category-I, OSEPA Upper Primary — Science, OSSC Mathematics, OSSSC Govt Physical Education Teacher (PET)

and 5 more exams

OSSSC Govt TGT Arts, OSSSC Govt TGT Science — CBZ, OSSSC Govt TGT Science — PCM, SSB Lecturer (Degree Colleges), SSB TGT PCM

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Overview

Standard-angle values, three identities and the tangent ratio for heights solve nearly all trigonometry items.

Key points

Standard values: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
Sin²θ + cos²θ = 1; sec²θ − tan²θ = 1; cosec²θ − cot²θ = 1.
Sin(90° − θ) = cos θ; tan(90° − θ) = cot θ.
Heights: tan θ = height ÷ horizontal distance.
Angle of elevation from the ground equals the angle of depression from the top (alternate angles).

Explanation

If sin θ = p/h, build the triangle: base = √(h² − p²).
For θ + φ = 90°, sin θ = cos φ and tan θ × tan φ = 1.
Two-angle height problems: write tan for each position and subtract distances.
Maximum of a sin θ + b cos θ is √(a² + b²).

Step by step

Draw the triangle.
Choose the ratio or identity.
Substitute standard values.

Examples

Shadow 20 m, angle of elevation of the Sun 45° → height = 20 × tan 45° = 20 m.
Angle 60°, shadow 10 m → height = 10√3 ≈ 17.32 m.
Sin θ = 3/5 → cos θ = 4/5, tan θ = 3/4.

Traps and advanced points

Trap: Using sin instead of tan for height and horizontal distance.
Trap: Mixing up the angle of elevation and the angle at the top.
Trap: Forgetting that sin²θ means (sin θ)².

Quick revision

Sin² + cos² = 1.
Tan = height/base.
Sin(90° − θ) = cos θ.

Practice MCQs

Find 44 × sin 45° (use the standard trigonometric value).
1. 37.34
2. 31.11
3. 24.89
4. 43.56
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Answer: (b) 31.11. Use standard trigonometric values.
Find 54 × sin 45° (use the standard trigonometric value).
1. 45.82
2. 53.46
3. 38.18
4. 30.55
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Answer: (c) 38.18. Use standard trigonometric values.
Find 64 × sin 45° (use the standard trigonometric value).
1. 54.31
2. 63.36
3. 36.2
4. 45.25
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Answer: (d) 45.25. Use standard trigonometric values.
Find 74 × sin 45° (use the standard trigonometric value).
1. 62.79
2. 52.33
3. 73.26
4. 41.86
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Answer: (b) 52.33. Use standard trigonometric values.
Find 84 × sin 45° (use the standard trigonometric value).
1. 71.28
2. 83.16
3. 47.52
4. 59.4
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Answer: (d) 59.4. Use standard trigonometric values.
Find 76 × sin 90° (use the standard trigonometric value).
1. 106.4
2. 76
3. 60.8
4. 91.2
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Answer: (b) 76. Use standard trigonometric values.
Find 38 × sin 30° (use the standard trigonometric value).
1. 19
2. 22.8
3. 15.2
4. 26.6
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Answer: (a) 19. Use standard trigonometric values.
Find 87 × sin 0° (use the standard trigonometric value).
1. 0
2. -1
3. 1
4. 2
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Answer: (a) 0. Use standard trigonometric values.
Find 61 × sin 90° (use the standard trigonometric value).
1. 61
2. 48.8
3. 85.4
4. 73.2
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Answer: (a) 61. Use standard trigonometric values.
Find 8 × sin 30° (use the standard trigonometric value).
1. 5
2. 6
3. 3
4. 4
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Answer: (d) 4. Use standard trigonometric values.
Find 11 × sin 90° (use the standard trigonometric value).
1. 13.2
2. 11
3. 15.4
4. 8.8
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Answer: (b) 11. Use standard trigonometric values.
Find 42 × sin 0° (use the standard trigonometric value).
1. 2
2. 0
3. -1
4. 1
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Answer: (b) 0. Use standard trigonometric values.
Find 99 × sin 45° (use the standard trigonometric value).
1. 98
2. 70
3. 84
4. 56
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Answer: (b) 70. Use standard trigonometric values.
Find 55 × sin 60° (use the standard trigonometric value).
1. 38.11
2. 66.68
3. 47.63
4. 57.16
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Answer: (c) 47.63. Use standard trigonometric values.
Find 14 × sin 45° (use the standard trigonometric value).
1. 9.9
2. 13.86
3. 11.88
4. 7.92
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Answer: (a) 9.9. Use standard trigonometric values.