Degrees, Radians and the Unit Circle

Definition A degree is $\frac{1}{360}$ of a full rotation, so a full turn measures $360^\circ$, a straight angle $180^\circ$ and a right angle $90^\circ$. The SAT reference sheet states that the number of degrees of arc in a circle is $360$.

Definition Place an angle with its vertex at the center of a circle of radius $r$. The angle measures one radian when the arc it cuts off has length $r$. In general the radian measure of a central angle is

where $s$ is the length of the arc it cuts off. A full turn cuts off the whole circumference $2\pi r$, so a full turn is $2\pi$ radians and

The reference sheet states that the number of radians of arc in a circle is $2\pi$. Because the ratio $\frac{s}{r}$ is the same for every circle with the same central angle, the radian measure depends only on the angle. When an angle is written with no degree sign, such as $\theta = 2$, it is in radians.

Definition On a circle of radius $r$, a central angle of $\theta$ radians cuts off an arc of length

This formula requires $\theta$ in radians. For an angle of $d$ degrees, use the fraction of the full circle instead,

The arc length is in the same unit as the radius.

Definition The unit circle is the circle of radius $1$ centered at the origin, the set of points $(x, y)$ with

An angle $\theta$ is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis. A positive angle rotates counterclockwise and a negative angle rotates clockwise. The terminal side of the angle meets the unit circle in exactly one point. For a positive angle, the arc swept on the unit circle from $(1, 0)$ to that point has length equal to the radian measure, since $s = 1 \cdot \theta$.

Drawn on a graph the unit circle looks round only when both axes use the same scale. With unequal scales it looks like an oval, though the equation is unchanged.

Definition For any angle $\theta$, let $(x, y)$ be the point where its terminal side in standard position meets the unit circle. Then

So the point is $(\cos \theta, \sin \theta)$, with the cosine first and the sine second. For an acute angle this agrees with the right triangle definitions, because dropping a perpendicular from the point to the $x$-axis makes a right triangle with hypotenuse $1$, adjacent leg $x$ and opposite leg $y$. It also gives the identity

Definition The axes split the plane into four quadrants, numbered counterclockwise starting from the upper right. Quadrant I has $x > 0, y > 0$, quadrant II has $x < 0, y > 0$, quadrant III has $x < 0, y < 0$ and quadrant IV has $x > 0, y < 0$. Points on an axis lie in no quadrant. The quadrant of an angle’s terminal side therefore fixes the signs of $\cos \theta$ and $\sin \theta$, and $\tan \theta = \frac{\sin \theta}{\cos \theta}$ when $\cos \theta \ne 0$.

An angle outside $0^\circ$ to $360^\circ$ ends where it would after adding or subtracting $360^\circ$ enough times, so $-30^\circ$ and $330^\circ$ both end in quadrant IV.