Convert between degrees and radians; use the unit circle to define the sine and cosine of any angle as coordinates of a point; evaluate the trig functions at special angles; and apply the Pythagorean identity.

What this topic is asking

The Regents Algebra II exam (the Trigonometric Functions, F-TF, cluster) wants you to convert between degrees and radians, use the unit circle to define the sine and cosine of any angle as the coordinates of a point, evaluate the trig functions at special angles, and apply the Pythagorean identity. The unit circle extends trigonometry beyond the right triangles of Geometry to all angles.

Radians and the conversion

A radian measures an angle by arc length: one radian is the angle whose arc equals the radius. Since the full circumference is $2\pi r$, a full turn is $2\pi$ radians.

So $90^\circ = \frac{\pi}{2}$, $60^\circ = \frac{\pi}{3}$, and $\frac{3\pi}{4}$ radians $= 135^\circ$. The reference sheet gives the relationship $1$ radian $= \frac{180}{\pi}$ degrees, but you must apply it in the correct direction.

The unit circle defines sine and cosine

On the unit circle (centered at the origin with radius 1), the point reached by rotating an angle $\theta$ from the positive x-axis has coordinates:

This defines cosine as the x-coordinate and sine as the y-coordinate for any angle, not just acute ones. The signs follow the quadrant: in Quadrant II, $x < 0$ and $y > 0$, so cosine is negative and sine is positive; in Quadrant III both are negative; in Quadrant IV cosine is positive and sine is negative.

Special angles and reference angles

The special angles $\frac{\pi}{6}$ (30 degrees), $\frac{\pi}{4}$ (45 degrees), and $\frac{\pi}{3}$ (60 degrees), together with the quadrantal angles, have exact coordinates worth knowing. A reference angle is the acute angle to the x-axis; it gives the magnitude of sine and cosine, and the quadrant supplies the sign.

The Pythagorean identity

Because the unit-circle point $(\cos\theta, \sin\theta)$ lies on $x^2 + y^2 = 1$, the coordinates satisfy:

This identity (not on the reference sheet) lets you find one of sine or cosine from the other. A clarifying point worth stressing is that the identity gives the magnitude, while the quadrant gives the sign: solving $\cos^2\theta = \frac{16}{25}$ yields $\cos\theta = \pm\frac{4}{5}$, and you choose the sign from the quadrant you are told. Skipping the quadrant reasoning, and so giving the wrong sign, is the most common error on these questions.

Try this

Q1. Convert $\frac{\pi}{4}$ radians to degrees. [1 credit]

• Cue. $\frac{\pi}{4} \times \frac{180}{\pi} = 45$ degrees.

Q2. If $\cos\theta = \frac{5}{13}$ and $\theta$ is in Quadrant I, find $\sin\theta$. [2 credits]

• Cue. $\sin^2\theta = 1 - \frac{25}{169} = \frac{144}{169}$, so $\sin\theta = \frac{12}{13}$ (positive in QI).