Trigonometry

Trigonometry desribes the relationships between the sides and angles of triangles.

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}$

• $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{c}{a}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c}$

• $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{c}{b}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}$
• $\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{b}{a}$

These definitions can be extended to obtuse angles, by changing the side lengths’ signs when they cross the starting point.

Important Angles

On the unit circle, $\sin \theta$ corresponds to $y$ values, and $\cos \theta$ corresponds to $x$ values.

• $\sin \frac{\pi}{6} = \cos \frac{\pi}{3} = \frac{1}{2}$
• $\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$
• $\sin \frac{\pi}{3} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$