Unit Circle

Key Facts

• Coordinates on unit circle: (cosθ, sinθ)
• sin²θ + cos²θ = 1 (Pythagorean identity, directly from unit circle)
• Conversion: 1 radian = 180/π degrees ≈ 57.3°
• Key values: $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$, $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
• Sine is an odd function: $\sin(-\theta) = -\sin\theta$. Cosine is even: $\cos(-\theta) = \cos\theta$.
• Period of sine and cosine is 2π radians (360°).

Vocabulary

Radian

Angle unit where 2π radians = 360°; the arc length on a unit circle equals the angle in radians.

Reference angle

The acute angle between the terminal side and the x-axis; used to find trig values in any quadrant.

Sine

The y-coordinate of the corresponding point on the unit circle.

Cosine

The x-coordinate of the corresponding point on the unit circle.

Tangent

$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$ on the unit circle; undefined when $\cos\theta = 0$.