Exact Values of Related Angles

Video lesson · from 4:566 worked examples

One lesson video covers all of Trigonometry, so it opens at 4:56 for this topic — not from the beginning.

The CAST diagram helps us determine the sign (positive or negative) of trigonometric functions in each of the four quadrants.

All +ve

Sine +ve

Tan +ve

Cos +ve

0 / 360°

90°

180°

270°

We can use this diagram along with our exact values in the first quadrant to evaluate trigonometric functions for any angle.

Find the acute related angle first: for $150^\circ$ that is $30^\circ$ . Work out the exact value for the acute angle, then attach the sign.
The quadrant decides the sign: use CAST. The size of the value never changes — only whether it is positive or negative.
Subtract from the right axis: related angles come from $180^\circ$ and $360^\circ$ for sine and cosine. Subtracting from $90^\circ$ changes the ratio instead of the sign.
Tangent has a period of $180^\circ$ : it repeats twice as often as sine and cosine, so treat it separately.

Example 1

Find the exact value of $\sin 150^\circ$ .

1. $150^\circ$ is in the 2nd quadrant, where Sine is positive.

2. The related acute angle is $180^\circ - 150^\circ = 30^\circ$ .

\begin{aligned} \sin 150^\circ &= \sin 30^\circ \\ &= \frac{1}{2} \end{aligned}

Example 2

Find the exact value of $\tan 120^\circ$ .

1. $120^\circ$ is in the 2nd quadrant, where Tangent is negative.

2. The related acute angle is $180^\circ - 120^\circ = 60^\circ$ .

\begin{aligned} \tan 120^\circ &= -\tan 60^\circ \\ &= -\sqrt{3} \end{aligned}

Example 3

Find the exact value of $\cos 405^\circ$ .

1. $405^\circ$ is more than one full circle ( $360^\circ$ ).

2. The equivalent angle is $405^\circ - 360^\circ = 45^\circ$ , which is in the 1st quadrant.

\begin{aligned} \cos 405^\circ &= \cos 45^\circ \\ &= \frac{1}{\sqrt{2}} \end{aligned}

Example 4

Find the exact value of $\cos\left(\frac{7\pi}{4}\right)$ .

1. $\frac{7\pi}{4}$ is in the 4th quadrant, where Cosine is positive.

2. The related acute angle is $2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$ .

\begin{aligned} \cos\left(\frac{7\pi}{4}\right) &= \cos\left(\frac{\pi}{4}\right) \\ &= \frac{1}{\sqrt{2}} \end{aligned}

Example 5

Find the exact value of $\tan\left(\frac{2\pi}{3}\right)$ .

1. $\frac{2\pi}{3}$ is in the 2nd quadrant, where Tangent is negative.

2. The related acute angle is $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$ .

\begin{aligned} \tan\left(\frac{2\pi}{3}\right) &= -\tan\left(\frac{\pi}{3}\right) \\ &= -\sqrt{3} \end{aligned}

Example 6

Find the exact value of $\cos\left(-\frac{2\pi}{3}\right)$ .

1. A negative angle means we go backwards (clockwise).

2. $-\frac{2\pi}{3}$ places us in the 3rd quadrant, where Cosine is negative.

3. The related acute angle is $\frac{\pi}{3}$ .

\begin{aligned} \cos\left(-\frac{2\pi}{3}\right) &= -\cos\left(\frac{\pi}{3}\right) \\ &= -\frac{1}{2} \end{aligned}