Learn · Mathematics · Trigonometry

The unit circle: sin, cos, tan and cot as lengths

Turn the angle and see where each value lives on the circle, which quadrant makes it positive, and why tan and cot are sometimes undefined.

High school Lesson 2 of 9 Interactive figure Practice

First, the rules

A circle of radius 1

Draw a circle of radius 1 around the origin. Measure an angle θ from the positive x-axis, counter-clockwise. The point P where the arm meets the circle carries every value.

★ The point on the circle

Examples 1, 2

P = (cos θ, sin θ)
cos θ is how far right P is, sin θ is how high.

Tangent and cotangent as lengths

Example 1

tan θ: where the arm (extended) crosses the vertical line x = 1.
cot θ: where it crosses the horizontal line y = 1.

Undefined values

At 90° and 270° the arm is parallel to x = 1 — tan is undefined. At 0° and 180° it is parallel to y = 1 — cot is undefined.

★ Signs by quadrant

Example 2

I: all positive · II: only sin · III: only tan and cot · IV: only cos
Counter-clockwise: "All Students Take Calculus".

Reference angle

Example 2

The acute angle between the arm and the x-axis. 150° → 30°, 225° → 45°, 300° → 60°. The value is the special-angle value; only the sign changes.

Radians

180° = π rad, so 30° = π6, 45° = π4, 60° = π3, 90° = π2

See it

Turn the angle

Use the buttons, the slider, or click a dot on the circle. Each coloured line is where that value lives; its length is the size of the value and its direction is the sign. Switch lines off to focus on one.

Step by step

Worked examples

|AB| in terms of θ

unit circle · cotangent

B is the point on the unit circle at angle θ. The horizontal line through B meets the y-axis at A. Express |AB| in terms of θ.

Ask yourself: BA is parallel to the x-axis. What is the angle at A? (Alternate angles: θ.)

Show the solution

Radius of the unit circle is 1, so |OB| = 1.
BA ∥ Ox, so the angle at A equals θ (alternate interior angles).
In the right triangle OBA, seen from A: |OB| is opposite, |AB| is adjacent.
cot θ = adjacentopposite = |AB|1

Answer |AB| = cot θ

Area of ABC in terms of α

unit circle · tangent

B(−1, 0) and C(1, 0) are the ends of a diameter of the unit circle. A is on the vertical line through C, and the angle ABC is α. Find the area of triangle ABC.

Ask yourself: What are the coordinates of B and C? How far apart are they?

Show the solution

B = (−1, 0) and C = (1, 0), so |BC| = 2.
ABC is right-angled at C (the line through C is vertical). From the angle at B: tan α = |AC||BC| = |AC|2, so |AC| = 2 tan α.
Area of a right triangle = half the product of the legs: 2 · 2 tan α2

Answer A(ABC) = 2 tan α

Practice: exact values on the circle

0 / 0

sin, cos, tan or cot of a special angle anywhere on the circle — pick the exact value. Wrong answers come with the reference-angle explanation; "Show it on the figure" turns the circle to that angle. Your best score is kept in this browser only.

About this lesson

An interactive unit circle: drag the angle from 0° to 360° and see sine as a height, cosine as a horizontal distance, tangent on the vertical line x = 1 and cotangent on the horizontal line y = 1. Learn the quadrant signs, reference angles and why tan 90° and cot 0° are undefined, then test yourself on the exact values of the special angles.

Part of Learn mathematics · unit Trigonometry. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.