Introduction to Trigonometry

Trigonometry is the study of relationships between angles and sides in triangles.

Right Triangle Trigonometry

In a right triangle, we define three primary ratios:

For an angle $\theta$:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$

SOH-CAH-TOA

A helpful mnemonic:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

Reciprocal Functions

$\csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}}$

$\sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$

$\cot\theta = \frac{1}{\tan\theta} = \frac{\text{adjacent}}{\text{opposite}}$

Special Angles

Angle	sin	cos	tan
0°	0	1	0
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45°	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60°	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	1	0	undefined

Example

In a right triangle, if the opposite side is 3 and the hypotenuse is 5:

$\sin\theta = \frac{3}{5} = 0.6$

Using Pythagorean theorem, adjacent = 4:

$\cos\theta = \frac{4}{5} = 0.8$ $\tan\theta = \frac{3}{4} = 0.75$