Trigonometric Identities
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| Trigonometric Identities | |
|---|---|
| Basic Definitions | |
| Definition of tangent | $ \tan \theta = \frac{\sin \theta}{\cos\theta} $ |
| Definition of cotangent | $ \cot \theta = \frac{\cos \theta}{\sin\theta} \ $ |
| Definition of secant | $ \sec \theta = \frac{1}{\cos \theta} \ $ |
| Definition of cosecant | $ \csc \theta = \frac{1}{\sin \theta} \ $ |
| Definition of versed sine (versine) | $ \text{versin } \theta = 1- \cos \theta \ $ |
| Definition of versed cosine (versine) | $ \text{vercosin } \theta = 1+ \cos \theta \ $ |
| Definition of coversed sine (coversine) | $ \text{coversin } \theta = \text{cvs } \theta = 1- \sin \theta \ $ |
| Definition of coversed cosine (covercosine) | $ \text{covercosin } \theta = 1+ \sin \theta \ $ |
| Definition of haversed sine (haversine) | $ \text{haversin } \theta = \frac{1- \cos \theta}{2} $ |
| Definition of haversed cosine (havercosine) | $ \text{havercosin } \theta = \frac{1+ \cos \theta}{2} $ |
| Definition of hacoversed sine (hacoversin) | $ \text{hacoversin } \theta = \frac{1 - \sin \theta}{2} $ |
| Definition of hacoversed cosine (hacovercosin) | $ \text{hacovercosin } \theta = \frac{1 + \sin \theta}{2} $ |
| Definition of exterior secant (exsec) | $ \text{exsec } \theta = \sec \theta - 1 \ $ |
| Definition of exterior cosecant (excosec) | $ \text{excosec } \theta = \csc \theta - 1 \ $ |
| Definition of chord (crd) | $ \text{crd } \theta = 2 \sin(\frac{\theta}{2}) $ |
| Pythagorean identity and other related identities | |
| Pythagorean identity | $ \cos^2 \theta+\sin^2 \theta =1 \ $ |
| $ \sin^2 \theta = 1-\cos^2 \theta \ $ | |
| $ \cos^2 \theta = 1-\sin^2 \theta \ $ | |
| $ \sec^2 \theta = 1+\tan^2 \theta \ $ | |
| $ \csc^2 \theta = 1+\cot^2 \theta \ $ | |
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| Half-Angle Formulas | |
| Half-angle for sine | $ \sin \frac{\theta}{2} = \pm \sqrt{ \frac{1-\cos \theta}{2} } \ $ |
| Half-angle for cosine | $ \cos \frac{\theta}{2} = \pm \sqrt{ \frac{1+\cos \theta}{2} } \ $ |
| Half-angle for tangent | $ \tan \frac{\theta}{2} = \csc \theta - \cot \theta \ $ |
| Half-angle for tangent | $ \tan \frac{\theta}{2} =\pm\sqrt{\frac{1-\cos \theta}{ 1+\cos \theta }} \ $ |
| Half-angle for tangent | $ \tan \frac{\theta}{2} =\frac{\sin \theta}{ 1+\cos \theta } \ $ |
| Half-angle for tangent | $ \tan \frac{\theta}{2} =\frac{1-\cos \theta}{ \sin \theta } \ $ |
| Half-angle for cotangent | $ \cot \frac{\theta}{2} = \csc \theta + \cot \theta $ |
| Half-angle for cotangent | $ \cot \frac{\theta}{2} = \frac{1 + \cos \theta}{\sin \theta} $ |
| Half-angle for cotangent | $ \cot \frac{\theta}{2} = \pm \sqrt{1 + \cos \theta \over 1 - \cos \theta} $ |
| Half-angle for cotangent | $ \cot \frac{\theta}{2} = \frac{\sin \theta}{1 - \cos \theta} $ |
| Double-Angle Formulas | |
| double-angle for sine | $ \sin 2 \theta = 2 \sin \theta \cos \theta \ $ |
| double-angle for sine | $ \sin 2 \theta = \frac{ 2 \tan \theta}{1+ \tan^2 \theta } \ $ |
| double-angle for cosine | $ \cos 2 \theta =\cos^2 \theta - \sin^2 \theta \ $ |
| double-angle for cosine | $ \cos 2 \theta =2 \cos^2 \theta - 1 \ $ |
| double-angle for cosine | $ \cos 2 \theta =1- 2 \sin^2 \theta \ $ |
| double-angle for cosine | $ \cos 2 \theta =\frac{1- \tan^2 \theta}{ 1+\tan^2 \theta } \ $ |
| double-angle for tangent | $ \tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\, $ |
| double-angle for cotangent | $ \cot 2\theta = \frac{\cot^2 \theta - 1}{2 \cot \theta}\, $ |
| Triple-Angle Formulas | |
| triple-angle for sine | $ \begin{align}\sin 3\theta & = 3 \cos^2\theta \sin\theta - \sin^3\theta \\ & = 3\sin\theta - 4\sin^3\theta \end{align} $ |
| triple-angle for cosine | $ \begin{align}\cos 3\theta & = \cos^3\theta - 3 \sin^2 \theta\cos \theta \\ & = 4 \cos^3\theta - 3 \cos\theta\end{align} $ |
| triple-angle for tangent | $ \tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta} $ |
| tripe-angle for cotangent | $ \cot 3\theta = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta} $ |
| Angle sum and difference identities | |
| Sine | $ \sin \left( \theta\pm \alpha \right)=\sin \theta \cos \alpha \pm \cos \theta \sin \alpha $ |
| Cosine | $ \cos \left(\theta\pm \alpha \right)= \cos \theta \cos \alpha \mp \sin \theta \sin \alpha $ |
| Tangent | $ \tan \left(\theta\pm \alpha \right)= \frac {\tan \theta \pm \tan \alpha}{1 \mp \tan \theta \tan \alpha} $ |
| Arcsine | $ \arcsin\alpha \pm \arcsin\beta = \arcsin(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2}) $ |
| Arccosine | $ \arccos\alpha \pm \arccos\beta = \arccos(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)}) $ |
| Arctangent | $ \arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right) $ |
