Sine of 105° and cosine of 165°

Problem

Compute $\sin(105°)$ and $\cos(165°)$ using the addition formulae.

Solution

We break each angle down as a sum of special angles.

For the sine, $105° = 60°+45°$: $\sin 105° = \sin 60°\cos 45° + \cos 60°\sin 45° = \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} + \frac{1}{2}\cdot\frac{\sqrt{2}}{2} = \boxed{\dfrac{\sqrt{6}+\sqrt{2}}{4}}.$

For the cosine, $165° = 120°+45°$ (with $\cos 120° = -\tfrac12$, $\sin 120° = \tfrac{\sqrt3}{2}$): $\cos 165° = \cos 120°\cos 45° - \sin 120°\sin 45° = -\frac{1}{2}\cdot\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} = \boxed{-\dfrac{\sqrt{6}+\sqrt{2}}{4}}.$

Links

Topics: Trigonometric formulae
Concepts: Addition formulae
Functions: Cosine · Sine
Methods: Addition formulae
Skills: Calculating · Using formulae
Exercise types: Computing expressions