Trigonometry • Math 1060

At Utah State University, Math 1060 covers Trigonometry. This page provides a summary of identities and topics used in Trigonometry.

Trig Topic Reviews

Coming Soon

Trig Graphing Tool

Use Calculator

Unit Circle

This is a downloadable copy of the unit circle

$The unit circle used to visualize properties of a triangle with radius of length one, randing from angles zero to 2 times pi.$

$ \displaystyle (x = \cos\theta, y = \sin\theta) $

$ \displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$

Conversions

Radians to Degrees: $ \displaystyle \times\frac{180}{\pi} $

Degrees to Radians: $ \displaystyle\times\frac{\pi}{180} $

Arc Length: $ \displaystyle s=r\theta $

Bounds for Inverse Trig Functions

$ \displaystyle \sin^{-1}(\theta) : \left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$

$ \displaystyle \cos^{-1}(\theta) : \left[0, \pi\right]$

$ \displaystyle \tan^{-1}(\theta) : \left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$

Trig Identities

Basic and Pythagorean

$ \displaystyle \sin(\theta) = \frac{1}{\csc (\theta)}$

$ \displaystyle \cos (\theta) = \frac{1}{\sec (\theta)}$

$ \displaystyle \tan (\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{1}{\cot(\theta)}$

$ \displaystyle \csc(\theta) = \frac{1}{\sin(\theta)}$

$ \displaystyle \sec (\theta) = \frac{1}{\cos (\theta)}$

$ \displaystyle \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}$

$ \displaystyle \sin^2(\theta)+ \cos^2(\theta) = 1$

$ \displaystyle \cot^2(\theta) + 1 = \csc^2(\theta)$

$ \displaystyle \cos(-\theta) = \cos(\theta)$

$ \displaystyle \tan^2(\theta) + 1 = \sec^2(\theta)$

$ \displaystyle \sin(-\theta) = -\sin(\theta)$

$ \displaystyle \tan(-\theta) = -\tan(\theta)$

Angle Sum and Difference

$ \displaystyle \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$

$ \displaystyle \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$

$ \displaystyle \tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

Double-Angle and Half-Angle

$ \displaystyle \sin(2\theta) = 2\sin(\theta)\cos(\theta)$

$ \displaystyle \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

$ \displaystyle \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)= 1-2\sin^2(\theta) = 2\cos^2(\theta)-1$

$ \displaystyle \sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$

$ \displaystyle \cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$

$ \displaystyle \tan(\frac{\theta}{2}) = \frac{\sin (\theta)}{1 + \cos(\theta)}= \frac{1 - \cos(\theta)}{\sin(\theta)}= \pm \sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$

Power Reduction and Sum to Product

$ \displaystyle \sin^2(\theta) = \frac{1}{2}[1 - \cos(2\theta)]$

$ \displaystyle \cos^2(\theta) = \frac{1}{2}[1 + \cos(2\theta)]$

$ \displaystyle \tan^2(\theta) = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$

$ \displaystyle \sin(\alpha) + \sin(\beta) = 2\sin(\frac{x + y}{2})\cos(\frac{\alpha-\beta}{2})$

$ \displaystyle \sin(\alpha) - \sin(\beta) = 2\cos(\frac{\alpha+\beta}{2})\sin(\frac{\alpha-\beta}{2})$

$ \displaystyle \cos(\alpha) + \cos(\beta) = 2\cos(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})$

$ \displaystyle \cos(\alpha) - \cos(\beta) = -2\sin(\frac{\alpha+\beta}{2})\sin(\frac{\alpha-\beta}{2})$