Trigonometry Function Identities

Vocabulary flashcards covering core trigonometric identities presented in the reference sheet.

Quotient Identities

Trigonometric identities that define tangent and cotangent in terms of sine and cosine: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

Reciprocal Identities

Trigonometric identities expressing functions as reciprocals of each other: $\sin(\theta) = \frac{1}{\csc(\theta)}$, $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\cos(\theta) = \frac{1}{\sec(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, $\tan(\theta) = \frac{1}{\cot(\theta)}$, and $\cot(\theta) = \frac{1}{\tan(\theta)}$.

Pythagorean Identities

Trigonometric identities derived from the Pythagorean theorem: $\sin^2(\theta) + \cos^2(\theta) = 1$, $\sec^2(\theta) - \tan^2(\theta) = 1$, and $\csc^2(\theta) - \cot^2(\theta) = 1$.

Even/Odd Identities

Trigonometric identities reflecting function symmetry under angle negation: $\sin(-\theta) = -\sin(\theta)$, $\cos(-\theta) = \cos(\theta)$, $\tan(-\theta) = -\tan(\theta)$, $\cot(-\theta) = -\cot(\theta)$, $\csc(-\theta) = -\csc(\theta)$, and $\sec(-\theta) = \sec(\theta)$.

Cofunction Identities

Identities relating trigonometric functions of complementary angles ($\frac{\pi}{2}\,\text{radians} = 90^\circ$): $\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)$, $\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)$, $\tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta)$, $\cot\left(\frac{\pi}{2} - \theta\right) = \tan(\theta)$, $\csc\left(\frac{\pi}{2} - \theta\right) = \sec(\theta)$, and $\sec\left(\frac{\pi}{2} - \theta\right) = \csc(\theta)$.

Double Angle Identities

Identities expressing trigonometric functions of twice an angle: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$, and $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$.

Half Angle Identities

Identities expressing squared trigonometric functions in terms of double-angle cosine expressions: $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$, $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$, and $\tan^2(\theta) = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$.

Sum/Difference Identities

Trigonometric identities evaluating functions for the sum or difference of two angles: $\sin(\theta \pm \phi) = \sin(\theta)\cos(\phi) \pm \cos(\theta)\sin(\phi)$, $\cos(\theta \pm \phi) = \cos(\theta)\cos(\phi) \mp \sin(\theta)\sin(\phi)$, and $\tan(\theta \pm \phi) = \frac{\tan(\theta) \pm \tan(\phi)}{1 \mp \tan(\theta)\tan(\phi)}$.

Sum to Product of Two Angles

Identities converting sums or differences of trigonometric functions into products: $\sin(\theta) + \sin(\phi) = 2\sin\left(\frac{\theta + \phi}{2}\right)\cos\left(\frac{\theta - \phi}{2}\right)$, $\sin(\theta) - \sin(\phi) = 2\cos\left(\frac{\theta + \phi}{2}\right)\sin\left(\frac{\theta - \phi}{2}\right)$, $\cos(\theta) + \cos(\phi) = 2\cos\left(\frac{\theta + \phi}{2}\right)\cos\left(\frac{\theta - \phi}{2}\right)$, and $\cos(\theta) - \cos(\phi) = -2\sin\left(\frac{\theta + \phi}{2}\right)\sin\left(\frac{\theta - \phi}{2}\right)$.

Product to Sum of Two Angles

Identities converting products of trigonometric functions into sums or differences: $\sin(\theta)\sin(\phi) = \frac{1}{2}\left[\cos(\theta - \phi) - \cos(\theta + \phi)\right]$, $\cos(\theta)\cos(\phi) = \frac{1}{2}\left[\cos(\theta - \phi) + \cos(\theta + \phi)\right]$, $\sin(\theta)\cos(\phi) = \frac{1}{2}\left[\sin(\theta + \phi) + \sin(\theta - \phi)\right]$, and $\cos(\theta)\sin(\phi) = \frac{1}{2}\left[\sin(\theta + \phi) - \sin(\theta - \phi)\right]$.