keywords: double-angle, triple-angle, angle sum

Trigonometric Identities

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 \$
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 $\arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right)$

## Alumni Liaison

Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett