INTRODUCTION TO TRIGNOMETRIC RATIOS
1. sinθ = P/H
2. cosθ = B/H
3. tanθ = P/B
4. cosecθ = H/P
5. secθ = H/B
6. cotθ = B/P
* Formulae of six trigonometric ratios in terms of trigonometric ratios
1. sinθ = 1 / cosecθ ⇒ cosecθ = 1/sinθ ⇒ sinθ . cosecθ = 1
2. cosθ = 1/secθ ⇒ secθ = 1/cosθ ⇒ cosθ . secθ = 1
3. tanθ = 1/cotθ ⇒ cotθ = 1/tanθ ⇒ tanθ. cotθ = 1
4. tanθ = sinθ/cosθ ⇒ cotθ = cosθ/sinθ
* Trigonometric Identities:-
I sin²θ+cos²θ = 1
1. sin²θ = 1-cos²θ
2. sinθ = √(1-cos²θ) ̅
3. cos²θ = 1-sin²θ
4. cosθ = √(1-sin²θ) ̅
II. sec²θ - tan²θ = 1
1. sec²θ = 1 + tan²θ
2. secθ = √(1+ tan²θ) ̅
3. tan²θ = sec²θ -1
4. tanθ = √(sec²θ -1) ̅
5. tan²θ - sec²θ = -1
6. (secθ + tanθ) (secθ - tanθ) = 1
7. (tanθ + secθ) (tanθ - secθ) = -1
Ill. cosec²θ - cot²θ = 1
1. cosec²θ = 1 + cot²θ
2. cosecθ = √(1+ cot²θ) ̅
3. cot²θ = cosec²θ - 1
4. cotθ = √(cosec²θ - 1) ̅
5. cot²θ - cosec²θ = -1
6. (cosecθ + cotθ) (cosecθ - cotθ) = 1
7. (cotθ + cosecθ) (cotθ - cosecθ) = -1
* Formulae of six trigonometric ratios of negative angles.
1. sin (-θ) = - sinθ
2. cos (-θ) = + cosθ
3. tan (-θ) = - tanθ
4. cosec (-θ) = - cosecθ
5. sec (-θ) = + secθ
6. cot (-θ) = - cotθ
* Values of six trigonometric ratios 0°, 30°, 45°, 60°, 90°
Ratio √(0/4) √(1/4) √(2/4) √(3/4) √(4/4)
00 300 450 600 900
Sin θ 0 1 / 2 1 / √2 √3 / 2 1
Cos θ 1 √3 / 2 1 / √2 1 / 2 0
Tan θ 0 1 / √3 1 √3 α
Cosec θ α 2 √2 2 / √3 1
Sec θ 1 2 / √3 √2 2 α
Cot θ α √3 1 1 / √3 0
* Combinations
1. sin 30° = cos60° = 1/2
2. sin 60° = cos 30° = √3/2
3. cosec 30° = sec 60° = 2
4. cosec 60° = sec 30° = 2/√3
5. tan 30° = cot 60° = 1/√3
6. tan 60° = cot 30° = √3
7. sin 45° = cos 45° = 1/√2
8. cosec 45° = sec 45° = √2
9. tan 45° = cot 45° = 1
10. sin 90° = cos 0° = 1
11. cosec 90° = sec 0° = 1
12. sin 0° = cos 90° = 0
13. tan 0° = cot 90° = 0
* Formulae of six trigonometric ratios in terms (90° - θ), (90° + θ), (180° - θ), (180° + θ), (270° - θ), (270° + θ), (360° - θ), (360° + θ)
A 11 II III III as IV IV I
Ratio (90°-θ) (90°+θ) (180°-θ) (180°+ θ) (270°-θ) (270°+θ) (360°-θ) (360'+θ)
Sin + cos θ + cos θ + sin θ sin θ - cos θ - cos θ - sin θ + sin θ
Cos + sin θ sin θ cos θ cos θ sin θ + sin θ + sin θ + cos θ
Tan + cot θ cot θ + tan θ + tan θ + tan θ + cot θ tan θ + tan θ
Cosec + sec θ + sec θ + cosec θ cosec θ sec θ sec θ cosec θ + cosec θ
Sec + cosec θ cosec θ sec θ sec θ cosec θ + cosec θ + sec θ + sec θ
Cot + tan θ tan θ cot θ + cot θ + tan θ tan θ cot θ + cot θ
Even Multiples Odd Multiples Quadrant Angles
sin → sin sin ↔ cos 90°, 180°, 270°,
cos → cos cosec ↔ sec 360°, 450°, 540°,
tan → tan tan ↔ cot 630°, 720°, ......etc
cosec → cosec
sec → sec
cot → cot
