Trig Identities Cheat Sheet (Unit Circle, Printable PDF)

About this resource

What's inside this resource

Eleven pages compile trigonometric formulas by topic rather than presenting practice problems: the six ratio definitions (sin, cos, tan, csc, sec, cot) given twice, once in terms of a right triangle's opposite/adjacent/hypotenuse and once generalized to any angle using x, y, and a unit radius, followed by the Law of Sines, Law of Cosines, and Law of Tangents. Later pages cover domains, ranges, and periods of each trig function, double-angle, half-angle, cofunction, sum-and-difference, product-to-sum, and sum-to-product identities, inverse trig function definitions and properties, and a full table of exact sine, cosine, and tangent values for every special angle from 0° to 360°. No worked examples, explanatory text, or practice problems appear anywhere in the file - it is formula reference only, meant to sit beside separately assigned problems rather than teach a concept on its own.

In the classroom

Teaching tips & learning objectives

Learning objectives

Apply the six trigonometric ratio definitions (opposite/adjacent/hypotenuse) to set up a right-triangle problem
Apply the Law of Sines or Law of Cosines to solve for an unknown side or angle in a non-right triangle
Use double-angle, half-angle, or sum-and-difference identities to rewrite or simplify a trigonometric expression
Recall exact sine, cosine, and tangent values for special angles (0°, 30°, 45°, 60°, 90°, and their reflections) without a calculator

Teaching tips

Distribute as a reference sheet during timed tests or homework rather than as a stand-alone lesson, since it defines formulas without step-by-step teaching and assumes prior instruction
Highlight or tab the specific formula families a current unit covers (e.g. only the Law of Sines/Cosines during a triangles unit) so students aren't scanning all roughly 15 formula groups at once
Pair the special-angles table with unit-circle instruction, since values are given in exact fraction/radical form (e.g. root-3 over 2) that students need to be able to read and simplify, not just locate

Skills covered

Formula selection — students locate the appropriate trigonometric ratio, identity, or triangle law on a reference page.
Trigonometric relationships — the sheets organize reciprocal, quotient, Pythagorean, double-angle, half-angle, and sum-and-difference identities.
Triangle problem setup — right-triangle ratios and the Laws of Sines and Cosines support setting up side-and-angle calculations.
Exact-value reference — students consult special-angle and unit-circle values written in exact forms.

Good to know

Questions teachers ask about this resource

The sheet gives two different definitions of sine, cosine, and tangent - one assuming the angle is between 0 and pi/2, and one for 'any angle.' Are these alternative definitions for different problem types?

Yes. The first set (opposite/adjacent/hypotenuse) applies to an acute angle inside a right triangle, while the second set (based on x, y, and a radius of 1) generalizes the same ratios to any angle on the unit circle - the sheet includes both because triangle trig and unit-circle trig use different setups for the same functions.