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Trigonometry Values Guide

Knowing the exact values of sine, cosine, and tangent for standard angles — 0°, 30°, 45°, 60°, and 90° — is one of the most practical skills in trigonometry. These values come directly from the unit circle: a circle of radius 1 centered at the origin, where every point (cos θ, sin θ) encodes the cosine and sine of the angle θ. Once you memorize the first-quadrant values, symmetry rules let you extend them to all four quadrants. This guide links to every trig-value and unit-circle resource on Mathwords, so you can study exact values, identities, and inverse functions in one place.

Key Concepts

Unit Circle

The unit circle is a circle with radius 1 centered at the origin, used to define sine and cosine for all angles.

Exact Values of Trig Functions

A table of exact sine, cosine, and tangent values for the standard angles 0°, 30°, 45°, 60°, and 90°.

SOHCAHTOA

A mnemonic for the three basic trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Sine

Sine is the trigonometric function that gives the ratio of the opposite side to the hypotenuse in a right triangle, or the y-coordinate on the unit circle.

Cosine

Cosine is the trigonometric function that gives the ratio of the adjacent side to the hypotenuse in a right triangle, or the x-coordinate on the unit circle.

Tangent

Tangent is the trigonometric function equal to sine divided by cosine, giving the ratio of opposite to adjacent in a right triangle.

Reference Angle

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis, used to find trig values in any quadrant.

Radian

A radian is the angle measure where the arc length equals the radius; there are 2π radians in a full circle (360°).

Special Angles

Special angles (30°, 45°, 60° and their radian equivalents) have exact, memorizable trig values derived from the 30-60-90 and 45-45-90 triangles.

Pythagorean Identities

The Pythagorean identities (sin²θ + cos²θ = 1 and its variants) connect the squares of trig functions and follow directly from the unit circle.

Reciprocal Identities

Reciprocal identities define cosecant, secant, and cotangent as the reciprocals of sine, cosine, and tangent, respectively.

Unit Circle Trig Definitions

The unit-circle definitions extend sine, cosine, and the other trig functions from right-triangle ratios to all real-number angles.

All Trigonometry Values Guide Terms (32)

Frequently Asked Questions

How do I memorize the trig values for special angles?
A common method is the 'finger trick' or the pattern table: for sine at 0°, 30°, 45°, 60°, 90°, the values are √0/2, √1/2, √2/2, √3/2, √4/2 — which simplify to 0, 1/2, √2/2, √3/2, and 1. Cosine follows the same pattern in reverse. You can also derive all values from the 45-45-90 triangle (sides 1, 1, √2) and the 30-60-90 triangle (sides 1, √3, 2).
What is the unit circle and why is it important?
The unit circle is a circle of radius 1 centered at the origin. It is important because every point on it has coordinates (cos θ, sin θ), which defines sine and cosine for all angles — not just those in right triangles. It also reveals the periodic, symmetric nature of trig functions and makes it easy to find values in all four quadrants using reference angles.
What are the exact trig values for 30°, 45°, and 60°?
At 30° (π/6): sin = 1/2, cos = √3/2, tan = √3/3. At 45° (π/4): sin = √2/2, cos = √2/2, tan = 1. At 60° (π/3): sin = √3/2, cos = 1/2, tan = √3. These values come from the special right triangles and are used constantly throughout trigonometry and calculus.

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