Trigonometry formulas

Right-Triangle Definitions

\( \sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

\( \cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

\( \tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}} \)

\( \csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}} \)

\( \sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}} \)

\( \cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}} \)

Reduction Formulas

\( \sin(-x) = -\sin(x) \)

\( \cos(-x) = \cos(x) \)

\( \sin\left(\frac{\pi}{2} – x\right) = \cos(x) \)

\( \cos\left(\frac{\pi}{2} – x\right) = \sin(x) \)

\( \sin\left(\frac{\pi}{2} + x\right) = \cos(x) \)

\( \cos\left(\frac{\pi}{2} + x\right) = -\sin(x) \)

\( \sin(\pi – x) = \sin(x) \)

\( \cos(\pi – x) = -\cos(x) \)

\( \sin(\pi + x) = -\sin(x) \)

\( \cos(\pi + x) = -\cos(x) \)

Basic Identities

\( \sin^2x + \cos^2x = 1 \)

\( \tan^2x + 1 = \frac{1}{\cos^2x} \)

\( \cot^2x + 1 = \frac{1}{\sin^2x} \)

Sum and Difference Formulas

\( \sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha \)

\( \sin(\alpha – \beta) = \sin\alpha \cdot \cos \beta – \sin \beta \cdot \cos\alpha \)

\( \cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta – \sin\alpha \cdot \cos\beta \)

\( \cos(\alpha – \beta) = \cos\alpha \cdot \cos \beta + \sin\alpha \cdot \cos\beta \)

\( \tan(\alpha + \beta) = \frac{ \tan\alpha + \tan\beta}{1 – \tan\alpha \cdot  \tan\beta } \)

\( \tan(\alpha – \beta) = \frac{ \tan\alpha – \tan\beta}{1 + \tan\alpha \cdot  \tan\beta } \)

Double Angle and Half Angle Formulas

\( \sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha \)

\( \cos(2\,\alpha) = \cos^2\alpha – \sin^2\alpha \)

\( \tan(2\,\alpha) = \frac{2\,\tan\alpha}{1 – \tan^2\alpha} \)

\( \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1-\cos\alpha}{2}} \)

\( \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}} \)

\( \tan \frac{\alpha}{2} = \frac{1 – \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 – \cos\alpha} \)

\( \tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos\alpha}{1 – \cos\alpha} } \)

Other Useful Trig Formulas

Law of sines

\( \frac{\sin\alpha}{\alpha} = \frac{\sin\beta}{\beta} = \frac{\sin\gamma}{\gamma} \)

Law of cosines

\begin{aligned}
a^2 = b^2 + c^2 – 2\cdot b\cdot c\cdot \cos\alpha \\
b^2 = a^2 + c^2 – 2\cdot a\cdot c\cdot \cos\beta \\
c^2 = a^2 + b^2 – 2\cdot a\cdot b\cdot \cos\gamma
\end{aligned}

Area of triangle

\( A = \frac{1}{2} a\,b\, \sin\gamma \)

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