Unit circle

High School

Definition

A standard circle with a radius of 1 unit. It is commonly used in trigonometry to make solving angles and trigonomic functions easier to use and understand. By evaluating Sin and Cos angles in the unit circle, we are able to determine all of the 6 trigonometry function ratios as real numbers.

Worked examples

\(\cos(0) = 1, \sin(0) = 0\)

At angle 0, the point on the unit circle is (1, 0), so cosine is the x-coordinate and sine is the y-coordinate.

\(\cos\left(\frac{\pi}{2}\right) = 0, \sin\left(\frac{\pi}{2}\right) = 1\)

At 90° (π/2 radians), the point is (0, 1), giving cosine = 0 and sine = 1.

\(\tan\left(\frac{\pi}{4}\right) = \frac{\sin(\pi/4)}{\cos(\pi/4)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1\)

Tangent is sine divided by cosine; at 45°, both equal √2/2, so the ratio is 1.

Common mistakes

\(\sin(30^\circ) = \frac{\sqrt{3}}{2}\) → \(\sin(30^\circ) = \frac{1}{2}\) Sine of 30° is 1/2; cosine of 30° is √3/2. Don't swap them.
The unit circle only works for angles between 0° and 90°. → The unit circle defines all six trig functions for any angle. It extends to all quadrants and even negative or greater-than-360° angles.
\(\cos^2(\theta) + \sin^2(\theta) = 1\) only when \(\theta = 0\) → \(\cos^2(\theta) + \sin^2(\theta) = 1\) for every angle This Pythagorean identity holds because the radius of the unit circle is always 1.

Where you'll use it next

You'll use the unit circle to solve trigonometric equations, graph sine and cosine functions, work with radians, and find exact values of trig functions in precalculus and calculus.

Found in 1 StudyPug lesson

Mastering the Unit Circle: Your Gateway to Trigonometry

Precalculus

Dive into the world of trigonometry with our comprehensive unit circle guide. Learn essential concepts, memorization techniques, and real-world applications to elevate your math skills.

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