8
Introduction to Trigonometry
Trigonometric ratios · Standard values · Identities
Trigonometry unit
12 marks
12 marks
Trigonometric ratios
| Ratio | = | Reciprocal | = |
|---|---|---|---|
| sin θ | PH | cosec θ | HP |
| cos θ | BH | sec θ | HB |
| tan θ | PB | cot θ | BP |
“Pandit Badri Prasad · Har Har Bole”
sin = P/H, cos = B/H, tan = P/B (read top/bottom pairs)
Reciprocal & quotient relations
sin θ = 1cosec θ
cos θ = 1sec θ
tan θ = 1cot θ
tan θ = sin θcos θ
cot θ = cos θsin θ
★ Table of standard values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 12 | 1√2 | √32 | 1 |
| cos θ | 1 | √32 | 1√2 | 12 | 0 |
| tan θ | 0 | 1√3 | 1 | √3 | Not defined |
| cosec θ | Not defined | 2 | √2 | 2√3 | 1 |
| sec θ | 1 | 2√3 | √2 | 2 | Not defined |
| cot θ | Not defined | √3 | 1 | 1√3 | 0 |
Trick to remember the table
sin row: write √0/4, √1/4, √2/4, √3/4, √4/4 → simplify.
cos row = sin row reversed. tan = sin ÷ cos.
cosec, sec, cot = reciprocals of sin, cos, tan.
Behaviour (0° → 90°)
- sin θ increases 0 → 1; cos θ decreases 1 → 0
- 0 ≤ sin θ ≤ 1, 0 ≤ cos θ ≤ 1
- sec θ ≥ 1, cosec θ ≥ 1 (never less than 1)
- sin 30° = cos 60°, tan 45° = 1
★ Trigonometric identities (true for all valid θ)
sin²θ + cos²θ = 1
sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
1 + tan²θ = sec²θ
sec²θ − tan²θ = 1
tan²θ = sec²θ − 1
1 + cot²θ = cosec²θ
cosec²θ − cot²θ = 1
cot²θ = cosec²θ − 1
1 + tan²θ = sec²θ is valid for 0° ≤ θ < 90°; 1 + cot²θ = cosec²θ for 0° < θ ≤ 90°.
Useful rearranged forms
sec θ − tan θ = 1sec θ + tan θ
cosec θ − cot θ = 1cosec θ + cot θ
Proving identities
- Start from the more complicated side.
- Convert everything to sin & cos if stuck.
- Multiply by the conjugate for terms like √(1 + sin θ)/(1 − sin θ).
- Use a² − b² = (a + b)(a − b).
Common mistakes
- sin θ is not sin × θ.
- sin²θ = (sin θ)², not sin(θ²).
- sin(A + B) ≠ sin A + sin B.
- Given one ratio (e.g. tan A = 34): draw the triangle, find the third side by Pythagoras.
Class 10 Maths · Complete Formula Sheet · CBSEBy Mandeep · Green Board
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