Trigonometry: exact values: solved examples

Worked examples of trigonometry: exact values, from easy to hard, with every step written out. Try each one yourself first, then check the working.

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Easy

Example 1. Find the exact value of $\cos 30^\circ$ .
1. Read the value from the special triangle or unit circle. $\cos 30^\circ = \frac{\sqrt{3}}{2}$
Answer: $\frac{\sqrt{3}}{2}$
Example 2. Find the exact value of $\sin 30^\circ$ .
1. Read the value from the special triangle or unit circle. $\sin 30^\circ = \frac{1}{2}$
Answer: $\frac{1}{2}$
Example 3. Find the exact value of $\cos 90^\circ$ .
1. Read the value from the special triangle or unit circle. $\cos 90^\circ = 0$
Answer: $0$

Example 4. Find the exact value of $\sin 90^\circ$ .
1. Read the value from the special triangle or unit circle. $\sin 90^\circ = 1$
Answer: $1$
Example 5. Find the exact value of $\cos 0^\circ$ .
1. Read the value from the special triangle or unit circle. $\cos 0^\circ = 1$
Answer: $1$
Example 6. Find the exact value of $\cos 45^\circ$ .
1. Read the value from the special triangle or unit circle. $\cos 45^\circ = \frac{\sqrt{2}}{2}$
Answer: $\frac{\sqrt{2}}{2}$

Example 7. Find the exact value of $\cos 150^\circ$ .
1. Read the value from the special triangle or unit circle. $\cos 150^\circ = \frac{-\sqrt{3}}{2}$
Answer: $\frac{-\sqrt{3}}{2}$
Example 8. Find the exact value of $\sin 150^\circ$ .
1. Read the value from the special triangle or unit circle. $\sin 150^\circ = \frac{1}{2}$
Answer: $\frac{1}{2}$
Example 9. Find the exact value of $\tan 330^\circ$ .
1. Read the value from the special triangle or unit circle. $\tan 330^\circ = \frac{-\sqrt{3}}{3}$
Answer: $\frac{-\sqrt{3}}{3}$

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