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.

It shows a periodic wave oscillating between -1 and 1, with x-intercepts at multiples of $\pi$.

The graph of cosine is a sine wave shifted $\frac{\pi}{2}$ units to the left. It starts at its maximum value.

It indicates a change in amplitude, making the wave taller or shorter.

It represents a circle centered on the x-axis passing through the origin.

It has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where n is an integer, and a period of $\pi$.

It's the inverse of the sine function, defined for [-1, 1] and ranging from $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

A circle centered at the origin with radius a.

The period decreases, causing the graph to oscillate more frequently.

Measure the distance between two consecutive peaks or troughs.

A spiral that extends outward from the origin as $\theta$ increases.

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.