# Theory : Trigonometry

## Trigonometric Identities and Formulas

There are three systems of measuring Angles-

### 1. Circular System

• Unit of Measurement is radian.
• 180 degree = π radians

### 2. Sexagesimal System (English System)

• Right angle is divided into 90 equal parts called degree.
• Unit of Measurement is degree.
• Each degree is divided into 60 equal parts called minute. (1 degree = 60’)
• Each minute divided into 60 equal parts called seconds (1 minute = 60’’)

### 3. Centesimal or French System

• Right angle is divided into 100 equal parts.
• Unit of measurement is grades.
• Each grade is divided into 100 equal parts called minute, and minutes into seconds.

Sign Conventions

• cos (90 – θ) = sinθ
• tan (90 – θ) = cotθ
• cosec (90 – θ) = secθ
• sec (90 – θ) = cosecθ
• cot (90 – θ) = tanθ

### Some Basic Formulas and Identities

1. sin2θ + cos2θ = 1
2. 1 + tan2θ = sec2θ
3. 1 + cot2θ = cosec2θ
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1. sin (A + B) = sin A × cos B + cos A × sin B
2. sin (A – B) = sin A × cos B – cos A × sin B
3. cos (A + B) = cos A × cos B – sin A × sin B
4. cos (A – B) = cos A × cos B + sin A × sin B
5. tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
6. tan (A – B) = (tan A – tan B) / (1+ tan A tan B)
7. cot (A + B) = (cot A cot B – 1) / (cot A + cot B)
8. cot (A – B) = (cot A cot B + 1) / (cot B – cot A)
9. sin 2θ = 2sinθcosθ
10. sin2θ = 2tanθ/1+tan2θ
11. cos2θ = 2cos2θ – 1
12. cos2θ = cos2θ – sin2θ
13. cos2θ = 1 – 2sin2θ
14. cos2θ = (1 – tan2θ)/( 1 + tan2θ)
15. tan2θ = 2tanθ/1 – tan2θ
16. sin3θ = 3sinθ – 4sin3θ
17. cos3θ = 4cos3θ – 3cosθ
18. tan3θ = (3tanθ – tan3θ)/(1 – 3tan2θ)
19. tan (A + B + C) = (tan A + tan B + tan C – tan A tan B tan C)/(1 – tan A tan B – tan B tan C – tan C tan A)
20. 2sin A sin B = cos (A – B) – cos (A + B)
21. 2cos A cos B = cos (A + B) + cos (A – B)
22. 2sin A cos B = sin (A + B) + sin (A – B)
23. 2cos A sin B = sin (A + B) – sin (A – B)
24. sinC + sinD = 2sin[(C + D)/2] × cos[(C – D)/2]
25. sinC – sinD = 2cos[(C + D)/2] × sin[(C – D)/2]
26. cosC + cosD = 2cos[(C + D)/2]cos[(C – D)/2]
27. cosC – cosD = 2sin[(C + D)/2]cos[(D – C)/2]
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