Trigonometry Ratios & Unit Circle Cheat Sheet

A printable reference covering sine, cosine, tangent, SOH-CAH-TOA, unit circle coordinates, radians, and reciprocal identities for grades 9-11.

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Trigonometry ratios connect angle measures to side lengths in right triangles and to coordinates on the unit circle. This cheat sheet helps students quickly choose the correct ratio, identify exact values, and move between degrees and radians. It is especially useful for solving triangle problems, graphing trig functions, and preparing for algebra, geometry, or precalculus assessments. The core ratios are $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ , often remembered with SOH-CAH-TOA. On the unit circle, each point has coordinates $(\cos \theta, \sin \theta)$ , which makes exact trig values easier to organize. Important ideas include reciprocal identities, quadrant signs, reference angles, and radian measure using $180^{\circ} = \pi$ radians.

Key Facts

SOH-CAH-TOA means $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ , $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ , and $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ .
The tangent ratio can also be written as $\tan \theta = \frac{\sin \theta}{\cos \theta}$ when $\cos \theta \neq 0$ .
On the unit circle, the point at angle $\theta$ is $(x,y) = (\cos \theta, \sin \theta)$ .
The Pythagorean identity is $\sin^2 \theta + \cos^2 \theta = 1$ .
The reciprocal identities are $\csc \theta = \frac{1}{\sin \theta}$ , $\sec \theta = \frac{1}{\cos \theta}$ , and $\cot \theta = \frac{1}{\tan \theta}$ .
To convert degrees to radians, multiply by $\frac{\pi}{180^{\circ}}$ , so $60^{\circ} = \frac{\pi}{3}$ .
To convert radians to degrees, multiply by $\frac{180^{\circ}}{\pi}$ , so $\frac{3\pi}{4} = 135^{\circ}$ .
Quadrant signs follow ASTC: in Quadrant I all are positive, in Quadrant II sine is positive, in Quadrant III tangent is positive, and in Quadrant IV cosine is positive.

Vocabulary

Sine: Sine is the ratio $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ in a right triangle and the $y$ -coordinate on the unit circle.
Cosine: Cosine is the ratio $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ in a right triangle and the $x$ -coordinate on the unit circle.
Tangent: Tangent is the ratio $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ and also equals $\frac{\sin \theta}{\cos \theta}$ when defined.
Unit Circle: The unit circle is a circle with radius $1$ centered at the origin, where each angle $\theta$ corresponds to $(\cos \theta, \sin \theta)$ .
Reference Angle: A reference angle is the acute angle between the terminal side of an angle and the $x$ -axis.
Radian: A radian is an angle measure based on arc length, where $180^{\circ} = \pi$ radians.

Common Mistakes to Avoid

Confusing opposite and adjacent sides is wrong because these sides depend on the chosen angle $\theta$ , not on the picture alone.
Using $\tan \theta = \frac{\text{adjacent}}{\text{opposite}}$ is wrong because tangent is defined as $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ .
Swapping unit circle coordinates is wrong because the correct ordered pair is $(\cos \theta, \sin \theta)$ , not $(\sin \theta, \cos \theta)$ .
Forgetting quadrant signs gives incorrect exact values because a reference angle only gives the size of the value, while the quadrant determines whether it is positive or negative.
Treating degrees and radians as the same unit is wrong because $60^{\circ}$ and $60$ radians are very different angle measures.

Practice Questions

1 In a right triangle, angle $\theta$ has opposite side $9$ and hypotenuse $15$ . Find $\sin \theta$ and the adjacent side length.
2 Convert $150^{\circ}$ to radians and find $\sin 150^{\circ}$ using the unit circle.
3 Find $\cos \frac{5\pi}{3}$ , $\sin \frac{5\pi}{3}$ , and $\tan \frac{5\pi}{3}$ .
4 Explain why $\sin \theta$ is positive but $\cos \theta$ is negative for an angle in Quadrant II.