Trigonometric Functions

Trigonometric functions link the measure of an angle to the lengths of segments projected onto the axes.

For any angle α, there is exactly one point P on the circumference, defined by its intersection with the radius.

an angle with measure alpha

The projection of point P onto the y-axis creates a segment OPy, known as the sine.

the sine function

Similarly, projecting point P onto the x-axis forms a segment OPx, called the cosine.

the cosine function

Other common trigonometric functions include the tangent and cotangent.

the tangent function

Key Angles

Below are some important angles for sine and cosine functions:

Angle (Degrees)	Angle (Radians)	Sine	Cosine	Tangent	Cotangent
$$ 0° $$	$$ 0 $$	$$ 0 $$	$$ 1 $$	$$ 0 $$	$$ \nexists $$
$$ 30° $$	$$ \frac{\pi}{6} $$	$$ \frac{1}{2} $$	$$ \frac{\sqrt{3}}{2} $$	$$ \frac{\sqrt{3}}{3} $$	$$ \sqrt{3} $$
$$ 45° $$	$$ \frac{\pi}{4} $$	$$ \frac{\sqrt{2}}{2} $$	$$ \frac{\sqrt{2}}{2} $$	$$ 1 $$	$$ 1 $$
$$ 60° $$	$$ \frac{\pi}{3} $$	$$ \frac{\sqrt{3}}{2} $$	$$ \frac{1}{2} $$	$$ \sqrt{3} $$	$$ \frac{\sqrt{3}}{3} $$
$$ 90° $$	$$ \frac{\pi}{2} $$	$$ 1 $$	$$ 0 $$	$$ \nexists $$	$$ 0 $$
$$ 180° $$	$$ \pi $$	$$ 0 $$	$$ -1 $$	$$ 0 $$	$$ \nexists $$
$$ 270° $$	$$ \frac{3 \pi}{2} $$	$$ -1 $$	$$ 0 $$	$$ \nexists $$	$$ 0 $$

The cosine is a periodic function with a period of 2π.

Trigonometric functions repeat at regular intervals, known as their periods.

Some functions share the same period length.

And so on.