Trigonometry Formulae

A complete collection of basic to advanced trigonometric identities and formulae — nripendraswar.com.np

I. Compound Angle Formulae

Definition: A compound angle is an angle formed by the algebraic sum or difference of two or more angles. If $A$ and $B$ are two distinct angles, then $(A + B)$ and $(A - B)$ are compound angles.

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. $\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$

6. $\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}$

7. $\cot(A + B) = \dfrac{\cot A \cot B - 1}{\cot B + \cot A}$

8. $\cot(A - B) = \dfrac{\cot A \cot B + 1}{\cot B - \cot A}$

9. $\sin(A + B) \sin(A - B) = \sin^2 A - \sin^2 B$

10. $\cos(A + B) \cos(A - B) = \cos^2 A - \sin^2 B$

11. $\sin(A + B) + \sin(A - B) = 2 \sin A \cos B$

12. $\sin(A + B) - \sin(A - B) = 2 \cos A \sin B$

13. $\cos(A + B) + \cos(A - B) = 2 \cos A \cos B$

14. $\cos(A + B) - \cos(A - B) = -2 \sin A \sin B$

II. Multiple Angle Formulae

Definition: A multiple angle is an angle formed by multiplying a given angle by a positive integer. If $A$ is an angle, then $2A, 3A, 4A, \dots$ are multiple angles. The formulae related to these angles are called multiple angle formulae.

1. $\sin 2A = 2 \sin A \cos A$

2. $\sin 2A = \dfrac{2 \tan A}{1 + \tan^2 A}$

3. $\cos 2A = \cos^2 A - \sin^2 A$

4. $\cos 2A = 2 \cos^2 A - 1$

5. $\cos 2A = 1 - 2 \sin^2 A$

6. $\cos 2A = \dfrac{1 - \tan^2 A}{1 + \tan^2 A}$

7. $\tan 2A = \dfrac{2 \tan A}{1 - \tan^2 A}$

8. $\sin 3A = 3 \sin A - 4 \sin^3 A$

9. $\cos 3A = 4 \cos^3 A - 3 \cos A$

10. $\tan 3A = \dfrac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

III. Sub-Multiple Angle Formulae

Definition: A sub-multiple angle is an angle formed by dividing a given angle by a positive integer. If $A$ is an angle, then $\frac{A}{2}, \frac{A}{3}, \frac{A}{4}, \dots$ are sub-multiple angles. The formulae related to these angles are called sub-multiple angle formulae.

1. $\sin A = 2 \sin \dfrac{A}{2} \cos \dfrac{A}{2}$

2. $\sin A = \dfrac{2 \tan \dfrac{A}{2}}{1 + \tan^2 \dfrac{A}{2}}$

3. $\cos A = \cos^2 \dfrac{A}{2} - \sin^2 \dfrac{A}{2}$

4. $\cos A = 2 \cos^2 \dfrac{A}{2} - 1$

5. $\cos A = 1 - 2 \sin^2 \dfrac{A}{2}$

6. $\cos A = \dfrac{1 - \tan^2 \dfrac{A}{2}}{1 + \tan^2 \dfrac{A}{2}}$

7. $\tan A = \dfrac{2 \tan \dfrac{A}{2}}{1 - \tan^2 \dfrac{A}{2}}$

8. $\sin A = 3 \sin \dfrac{A}{3} - 4 \sin^3 \dfrac{A}{3}$

9. $\cos A = 4 \cos^3 \dfrac{A}{3} - 3 \cos \dfrac{A}{3}$

10. $\tan A = \dfrac{3 \tan \dfrac{A}{3} - \tan^3 \dfrac{A}{3}}{1 - 3 \tan^2 \dfrac{A}{3}}$

IV. Quick or Speedy Formulae

1. $\sin^2 A = \dfrac{1 - \cos 2A}{2}$

2. $\cos^2 A = \dfrac{1 + \cos 2A}{2}$

3. $\sin^3 A = \dfrac{3 \sin A - \sin 3A}{4}$

4. $\cos^3 A = \dfrac{3 \cos A + \cos 3A}{4}$

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