2.1 · Angles, the unit circle, and exact values

Radians compare arc length with radius

An angle in radians satisfies $\displaystyle \theta=s/r$, so an arc has length $\displaystyle s=r\theta$. A full revolution is $\displaystyle 2\pi$ radians or $\displaystyle 360^\circ$. Convert degrees by multiplying by $\displaystyle \pi/180$.

The formula $\displaystyle s=r\theta$ requires radians. A radian is a dimensionless ratio, so arc length has the same units as the radius.

Coordinates define the circular functions

On the unit circle, the point at angle $\displaystyle \theta$ is $\displaystyle (\cos\theta,\sin\theta)$. Tangent is the ratio $\displaystyle \sin\theta/\cos\theta$ where cosine is nonzero. The reciprocal functions require their corresponding denominators to be nonzero.

Special triangles give the values for $\displaystyle \pi/6$, $\displaystyle \pi/4$, and $\displaystyle \pi/3$. Use the reference angle for magnitudes, then the quadrant for signs; do not memorize every signed value independently.

$$\cos^2\theta+\sin^2\theta=1$$

Coterminal angles describe the same point

Adding $\displaystyle 2\pi k$ for an integer $\displaystyle k$ leaves the unit-circle point unchanged. A reference angle is the acute angle between the terminal side and the horizontal axis.

At $\displaystyle 5\pi/6$, the reference angle is $\displaystyle \pi/6$ in Quadrant II, so cosine is negative and sine is positive. Reciprocal functions inherit the signs of their denominators.