The Unit Circle

Unit Circle Components

Angles are always expressed from the positive xx-axis, travelling in an anti-clockwise direction.

*A negative angle travels from the same position but in a clockwise direction.

The quadrants are numbered using I, II, III, IV.

*Angles can be larger than 360°, and really there is no limit. It just begins a new cycle.

The Unit Circle

The sine of an angle is the vertical component of a ray.

The cosine of an angle is the horizontal component of a ray.

The tangent of an angle is the length a ray reaches along the tangent x=1.

The trigonometric ratios become:

sin⁡θ=oppositehypotenuse=y1=y\sin\theta=\frac{opposite}{hypotenuse}=\frac{y}{1}=y

cos⁡θ=adjacenthypotenuse=x1=x\cos\theta=\frac{adjacent}{hypotenuse}=\frac{x}{1}=x

tan⁡θ=oppositeadjacent=yx\tan\theta=\frac{opposite}{adjacent}=\frac{y}{x}

The Other Quadrants