Trigonometric Formulas and Identities
\( \)\( \)\( \)Trigonometric Functions of Angle Sums/Differences
- \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
- \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
- \( \tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)
Sums of Trigonometric Functions into Products of Trigonometric Functions
- \( \sin A + \sin B = 2 \sin \left( \dfrac {A+B}{2} \right) \cos \left( \dfrac {A-B}{2} \right) \)
- \( \sin A - \sin B = 2 \sin \left( \dfrac {A - B}{2} \right) \cos \left( \dfrac {A + B}{2} \right) \)
- \( \cos A + \cos B = 2 \cos \left( \dfrac {A+B}{2} \right) \cos \left( \dfrac {A-B}{2} \right) \)
- \( \cos A - \cos B = - 2 \sin \left( \dfrac {A+B}{2} \right) \sin \left( \dfrac {A-B}{2} \right) \)
Product of Trigonometric Functions into Sum of Trigonometric Functions
- \( \sin A \, \cos B = \dfrac{1}{2} \; [ \sin(A+B) + \sin(A-B) ] \)
- \( \cos A \, \cos B = \dfrac{1}{2} \; [ \cos(A+B) + \cos(A-B) ] \)
- \( \sin A \, \sin B = - \dfrac{1}{2} \; [ \cos(A+B) - \cos(A-B) ] \)
Trigonometric Functions of Multiple Angles
- \( \sin (2 A) = 2 \sin A \cos A \)
- \( \cos (2 A) = 1 - 2 \sin^2 A = 2 \cos^2 A - 1 \)
- \( \sin (3 A) = 3 \sin A - 4 \sin^3 A \)
- \( \cos (3 A) = 4\cos^3 A - 3 \cos A \)
Power Reduction Identities
- \( \sin^2 A = \dfrac{1}{2} [1 - \cos (2A)] \)
- \( \cos^2 A = \dfrac{1}{2} [1 + \cos (2A)] \)
Half-Angle Formulas
- \( \sin (A/2) = \pm \sqrt {\dfrac{1- \cos A}{2}} \)
- \( \cos (A/2) = \pm \sqrt {\dfrac{1 + \cos A}{2}} \)
- \( \tan (A/2) = \dfrac{1 - \cos A}{\sin A} = \dfrac{\sin A}{1+\cos A} \)
Derivatives of Trigonometric Functions and Their Inverses
\( f(x) \) | \( \dfrac{d f(x)}{dx} \) |
|---|---|
| \( \sin x \) | \( \cos x \) |
| \( \cos x \) | \( - \sin x \) |
| \( \tan x \) | \( \sec^2 x \) |
| \( \cot x \) | \( - \csc^2 x \) |
| \( \sec x \) | \( \sec x \tan x \) |
| \( \csc x \) | \( - \csc x \cot x\) |
| \( \sin^{-1} x\) | \( \dfrac{1}{\sqrt{1-x^2}} \) |
| \( \cos^{-1} x\) | \( - \dfrac{1}{\sqrt{1-x^2}} \) |
| \( \tan^{-1} x\) | \( \dfrac{1}{1+x^2} \) |
Indefinite Integrals of Trigonometric Functions
Note that in all cases of indefinite integrals, the constant of integration is omitted here but should be added if necessary.
\( f(x) \) | \( \displaystyle \int f(x) dx \) |
|---|---|
| \( \sin x \) | \( -\cos x \) |
| \( \cos x \) | \( \sin x \) |
| \( \tan x \) | \( -\ln |\cos x| \) |
| \( \cot x \) | \( \ln |\sin x| \) |
| \( \sec x \) | \( \ln( \sec x + \tan x ) \) |
| \( \csc x \) | \( \ln(\csc x - \cot x) \) |
More References and Links
Handbook of Mathematical Functions
Mathematics for Engineers with Examples and Solutions