The Unit Circle: Beyond 90 Degrees

The right triangle works well for angles between 0° and 90°, but it cannot represent, for example, sin 150° or cos(−45°). The unit circle solves this problem.

The unit circle is a circle of radius 1 centered at the origin of a Cartesian coordinate system. For any angle θ — positive, negative, or greater than 360° — we draw a ray from the origin forming angle θ with the positive x-axis (counterclockwise for positive θ). The point P where this ray meets the circle has coordinates (cos θ, sin θ).

This definition is consistent with the right-triangle definition in the first quadrant (0° to 90°) and extends it naturally to all angles. Sine becomes the y-coordinate, cosine becomes the x-coordinate, and tangent continues to be sin/cos = y/x.

The Cartesian plane is divided into four quadrants. The sign of each trigonometric ratio depends on the quadrant in which the angle lies:

1st quadrant (0° to 90°): x > 0 and y > 0, so cos > 0, sin > 0, tan > 0. 2nd quadrant (90° to 180°): x < 0 and y > 0, so cos < 0, sin > 0, tan < 0. 3rd quadrant (180° to 270°): x < 0 and y < 0, so cos < 0, sin < 0, tan > 0. 4th quadrant (270° to 360°): x > 0 and y < 0, so cos > 0, sin < 0, tan < 0.

A popular mnemonic is 'All Students Take Calculus', reading the quadrants 1→4: All (all positive), Students (only sine positive), Take (only cosine positive), Calculus (only tangent positive).

To calculate the sine or cosine of any angle, find its reference angle — the acute angle formed with the nearest x-axis — and adjust the sign according to the quadrant.

Example: sin 150°. The angle is in the 2nd quadrant. The reference angle is 180° − 150° = 30°. In the 2nd quadrant, sine is positive. Therefore sin 150° = sin 30° = 1/2.

Example: cos 240°. The angle is in the 3rd quadrant. The reference angle is 240° − 180° = 60°. In the 3rd quadrant, cosine is negative. Therefore cos 240° = −cos 60° = −1/2.

This technique reduces any angle to one of the special angles (30°, 45°, 60°) or trivial ones (0°, 90°) that you already know.

After one full revolution (360° or 2π radians), the point on the circle returns to the same position. That is why sine and cosine have period 2π: sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ.

Tangent has period π (180°), since tan(θ + π) = tan θ. This happens because diametrically opposite points on the circle have opposite x and y coordinates, but the ratio y/x stays the same.

Periodicity is fundamental to the graphical representation of trigonometric functions — sinusoidal waves — and explains why they describe oscillatory phenomena such as sound, light, and alternating current.

Since the point (cos θ, sin θ) always lies on the circle of radius 1, its coordinates satisfy the circle equation: x² + y² = 1. Substituting gives: cos²θ + sin²θ = 1.

This is the most important trigonometric identity of all. From it, others follow: dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives cot²θ + 1 = csc²θ.

In practice, the fundamental identity is used to simplify expressions and solve equations. If you know sin θ = 3/5, you can immediately calculate cos θ: cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5 (the sign depends on the quadrant).