$\sin(\theta) = \frac{y}{r}$

$\cos(\theta) = \frac{x}{r}$

$\tan(\theta) = \frac{y}{x}$

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

$\text{radians} = \text{degrees} \times \frac{\pi}{180}$

$\text{degrees} = \text{radians} \times \frac{180}{\pi}$

$\sin(2x) = 2\sin(x)\cos(x)$

$\sin^2(\theta) + \cos^2(\theta) = 1$

$2\pi$

$\pi$

Trigonometric functions repeat their values at regular intervals. Sine and cosine repeat every 2π radians, while tangent repeats every π radians.

On the unit circle, sin(θ) is the y-coordinate and cos(θ) is the x-coordinate of a point on the circle corresponding to the angle θ.

Radians are based on the geometry of the circle (arc length to radius ratio), making them more natural in mathematical and physical contexts compared to the arbitrary division of degrees.

tan(θ) is undefined when cos(θ) = 0, which occurs at odd multiples of π/2.

Sine increases from 0 to 1 as the angle increases from 0 to π/2.

Cosine decreases from 1 to 0 as the angle increases from 0 to π/2.

Tangent increases from 0 to infinity as the angle increases from 0 to π/2.

Sine is positive in quadrants I and II, and negative in quadrants III and IV.

Cosine is positive in quadrants I and IV, and negative in quadrants II and III.

Tangent is positive in quadrants I and III, and negative in quadrants II and IV.

The derivative, cos(x), represents the slope of the sine function. Where sine increases, cosine is positive; where sine decreases, cosine is negative; where sine has a max/min, cosine is zero.

The derivative, -sin(x), represents the slope of the cosine function. Where cosine increases, -sine is positive; where cosine decreases, -sine is negative; where cosine has a max/min, -sine is zero.

The derivative, sec²(x), is always positive, indicating that the tangent function is always increasing (except where it's undefined).

The amplitude determines the maximum and minimum values of the function. A larger amplitude stretches the graph vertically.

The period determines how often the function repeats. A shorter period compresses the graph horizontally, while a longer period stretches it.

A vertical shift indicates a change in the midline of the function. The entire graph is shifted up or down.

A horizontal shift (phase shift) indicates a change in the starting point of the function. The entire graph is shifted left or right.

The period is the distance along the x-axis required for the function to complete one full cycle.

The amplitude is half the distance between the maximum and minimum values of the function.

The graph of sin(2x) is a horizontal compression of sin(x) by a factor of 2, meaning it oscillates twice as fast.