Trigonometry / Unit 3: The Unit Circle and Trig Functions of Any Angle
3.4 Reference Angles and Exact Values
Notes
To evaluate a trig function of any angle: find the reference angle, take the value for that acute angle, then apply the sign of the quadrant.
- Example: $\sin 240^\circ$. Reference angle $60^\circ$, Quadrant III so sine is negative: $\displaystyle -\frac{\sqrt{3}}{2}$.
- Negative angles turn clockwise: $\sin(-\theta) = -\sin\theta$ and $\cos(-\theta) = \cos\theta$.
- Angles larger than $360^\circ$ (or $2\pi$): subtract full turns first.
Practice
What is the exact value of $\cos 210^\circ$?
Reference angle $30^\circ$, Quadrant III, where cosine is negative.
What is the exact value of $\displaystyle \tan\frac{7\pi}{4}$?
Reference angle $\displaystyle \frac{\pi}{4}$, Quadrant IV, where tangent is negative.
What is the exact value of $\displaystyle \sin\left(-\frac{\pi}{6}\right)$?
Sine is odd: $\displaystyle \sin\left(-\frac{\pi}{6}\right) = -\sin\frac{\pi}{6} = -\frac{1}{2}$.
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