Patterns that repeat over time.

Periodic functions represented by sine or cosine.

Functions that find angles from given sine, cosine, or tangent values.

Another way to express positions in a plane using distance and angle.

The maximum displacement from the midline of the function.

The length of one complete cycle of the function.

A transformation that moves the graph of a function up or down.

A circle with radius 1 centered at the origin, used to define trigonometric functions.

A unit of angular measure defined as the angle subtended at the center of a circle by an arc equal in length to the radius.

A function defined in polar coordinates, typically in the form r = f(θ).

Amplitude: |A|, Period: $\frac{2\pi}{|B|}$

Find the principal value using $x = \arcsin(a)$, then use the properties of sine to find other solutions within the desired interval.

1. Calculate r: $r = \sqrt{3^2 + 4^2} = 5$. 2. Calculate $\theta$: $\theta = \arctan(\frac{4}{3}) \approx 0.93$ radians.

Set $\cos(2x) = 0$, solve for 2x: $2x = \frac{\pi}{2} + n\pi$, then solve for x: $x = \frac{\pi}{4} + \frac{n\pi}{2}$, where n is an integer.

Create a table of values for $\theta$ and r, plot the points (r, $\theta$), and connect them to form the graph (usually a circle).

Use trigonometric identities to simplify the equation, then solve for the multiple angle, and finally solve for the variable.

Identify the midline of the function. The vertical shift is the distance between the midline and the x-axis.

For $y = A\tan(Bx + C) + D$, the period is $\frac{\pi}{|B|}$.

Consider the range of the original trigonometric function. For example, the domain of $\arcsin(x)$ is [-1, 1].

Replace x with $r\cos(\theta)$ and y with $r\sin(\theta)$, then simplify the equation.

Sine and cosine are cofunctions, meaning $\sin(x) = \cos(\frac{\pi}{2} - x)$ and vice versa. They are also phase-shifted versions of each other.

Amplitude determines the maximum displacement of the function from its midline. A larger amplitude means a taller graph.

Period determines the length of one complete cycle. A smaller period means the function oscillates more rapidly.

Vertical shifts move the graph up or down, horizontal shifts move it left or right, amplitude changes stretch or compress it vertically, and period changes stretch or compress it horizontally.

To ensure they are functions (pass the vertical line test), their domains are restricted to principal values.

Use the formulas $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan(\frac{y}{x})$, considering the quadrant of (x, y).

The tangent of an angle is equal to the slope of a line making that angle with the x-axis.

Phase shift is a horizontal translation of the sinusoidal function, affecting its starting point.

The unit circle provides a visual representation of trigonometric functions, allowing for easy determination of values at special angles.

Polar coordinates are useful for representing circular or spiral paths and simplifying equations with circular symmetry.