Trigonometry has a wide set of identities that simplify calculations and problem-solving. Here are the most important formulae, grouped by type, for your reference.
1. Trigonometric Functions of Sum and Difference of Two Angles
\sin(A + B) = \sin A \cos B + \cos A \sin B
\sin(A - B) = \sin A \cos B - \cos A \sin B
\cos(A + B) = \cos A \cos B - \sin A \sin B
\cos(A - B) = \cos A \cos B + \sin A \sin B
\tan(A + B) = \frac{ \tan A + \tan B }{ 1 - \tan A \tan B }
\tan(A - B) = \frac{ \tan A - \tan B }{ 1 + \tan A \tan B }
\cot(A+B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}
\cot(A-B) = \frac{\cot A \cot B + 1}{\cot B - \cot A}
2. Product to Sum (or Product to Series) Formula
2\sin A \sin B = \cos(A - B) - \cos(A + B)
2\cos A \cos B = \cos(A + B) + \cos(A - B)
2\sin A \cos B = \sin(A + B) + \sin(A - B)
3. Sum to Product (or Series to Product) Formulas
\sin C + \sin D = 2 \sin\left( \frac{C + D}{2} \right) \cos\left( \frac{C - D}{2} \right)
\sin C - \sin D = 2 \cos\left( \frac{C + D}{2} \right) \sin\left( \frac{C - D}{2} \right)
\cos C + \cos D = 2 \cos\left( \frac{C + D}{2} \right) \cos\left( \frac{C - D}{2} \right)
\cos C - \cos D = -2 \sin\left( \frac{C + D}{2} \right) \sin\left( \frac{C - D}{2} \right)
4. Double Angle Formulas
\sin(2\theta) = 2 \sin\theta \cos\theta
\sin(2\theta) = \frac{2\tan\theta}{1 + \tan^2\theta}
\cos(2\theta) = \cos^2\theta - \sin^2\theta
\cos(2\theta) = 2\cos^2\theta - 1
\cos(2\theta) = 1 - 2\sin^2\theta
\cos(2\theta) = \frac{1 - \tan^2\theta}{1 + \tan^2\theta}
\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}
5. Triple Angle Formulas
\sin(3\theta) = 3\sin\theta - 4\sin^3\theta
\cos(3\theta) = 4\cos^3\theta - 3\cos\theta
\tan(3\theta) = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}
6. Other Commonly Used Trigonometric Identities
Basic Pythagorean Identities:
\sin^2\theta + \cos^2\theta = 1
1 + \tan^2\theta = \sec^2\theta
1 + \cot^2\theta = \csc^2\theta
Negative Angle Formulas:
\sin(-\theta) = -\sin\theta
\cos(-\theta) = \cos\theta
\tan(-\theta) = -\tan\theta
Co-function Identities:
\sin\left(90^\circ - \theta\right) = \cos\theta
\cos\left(90^\circ - \theta\right) = \sin\theta
\tan\left(90^\circ - \theta\right) = \cot\theta
Summary Table
| Category | Formula |
|---|---|
| Sine of Sum | \sin(A+B) = \sin A \cos B + \cos A \sin B |
| Sine of Difference | \sin(A-B) = \sin A \cos B - \cos A \sin B |
| Cosine of Sum | \cos(A+B) = \cos A \cos B - \sin A \sin B |
| Cosine of Diff. | \cos(A-B) = \cos A \cos B + \sin A \sin B |
| Tan of Sum | \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} |
| Tan of Diff. | \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} |
| Cot of Sum | \cot(A+B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} |
| Cot of Diff | \cot(A-B) = \frac{\cot A \cot B + 1}{\cot B - \cot A} |
| Product to Sum 1 | 2\sin A \sin B = \cos(A-B) - \cos(A+B) |
| Product to Sum 2 | 2\cos A \cos B = \cos(A+B) + \cos(A-B) |
| Product to Sum 3 | 2\sin A \cos B = \sin(A+B) + \sin(A-B) |
| Sum to Product 1 | \sin C + \sin D = 2 \sin\left( \frac{C + D}{2} \right) \cos\left( \frac{C - D}{2} \right) |
| Sum to Product 2 | \sin C - \sin D = 2 \cos\left( \frac{C + D}{2} \right) \sin\left( \frac{C - D}{2} \right) |
| Sum to Product 3 | \cos C + \cos D = 2 \cos\left( \frac{C + D}{2} \right) \cos\left( \frac{C - D}{2} \right) |
| Sum to Product 4 | \cos C - \cos D = -2 \sin\left( \frac{C + D}{2} \right) \sin\left( \frac{C - D}{2} \right) |
| Double Angle 1 | \sin(2\theta) = 2\sin\theta\cos\theta |
| Double Angle 2 | \sin(2\theta) = \frac{2\tan\theta}{1 + \tan^2\theta} |
| Double Angle 3 | \cos(2\theta) = \cos^2\theta - \sin^2\theta |
| Double Angle 4 | \cos(2\theta) = 2\cos^2\theta - 1 |
| Double Angle 5 | \cos(2\theta) = 1 - 2\sin^2\theta |
| Double Angle 6 | \cos(2\theta) = \frac{1 - \tan^2\theta}{1 + \tan^2\theta} |
| Double Angle 7 | \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta} |
| Triple Angle 1 | \sin(3\theta) = 3\sin\theta - 4\sin^3\theta |
| Triple Angle 2 | \cos(3\theta) = 4\cos^3\theta - 3\cos\theta |
| Triple Angle 3 | \tan(3\theta) = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} |
| Pythagorean 1 | \sin^2\theta + \cos^2\theta = 1 |
| Pythagorean 2 | 1 + \tan^2\theta = \sec^2\theta |
| Pythagorean 3 | 1 + \cot^2\theta = \csc^2\theta |
| Negative Angle 1 | \sin(-\theta) = -\sin\theta |
| Negative Angle 2 | \cos(-\theta) = \cos\theta |
| Negative Angle 3 | \tan(-\theta) = -\tan\theta |
| Co-function 1 | \sin(90^\circ - \theta) = \cos\theta |
| Co-function 2 | \cos(90^\circ - \theta) = \sin\theta |
| Co-function 3 | \tan(90^\circ - \theta) = \cot\theta |
Tip:
It is highly recommended to practice these formulae regularly, as they will make trigonometry questions much faster and easier to solve.