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

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

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

$\csc(\theta) = \frac{1}{\sin(\theta)}$

$\sec(\theta) = \frac{1}{\cos(\theta)}$

$\cot(\theta) = \frac{1}{\tan(\theta)}$

$x = r \cos(\theta)$

$y = r \sin(\theta)$

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

$\frac{2\pi}{|B|}$

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.

Sine: Starts at 0, odd function | Cosine: Starts at 1, even function

Polar: Uses radius and angle, good for circles | Rectangular: Uses x and y, good for lines

$\sin^{-1}(x)$: Inverse sine, returns an angle | $\frac{1}{\sin(x)}$: Cosecant, reciprocal of sine

Amplitude: Vertical distance from midline to max/min | Period: Horizontal length of one cycle

Phase Shift: Horizontal translation | Vertical Shift: Vertical translation

Tangent: $\frac{\sin(x)}{\cos(x)}$, asymptotes at $\frac{\pi}{2} + n\pi$ | Cotangent: $\frac{\cos(x)}{\sin(x)}$, asymptotes at $n\pi$

Secant: Reciprocal of cosine, even function | Cosecant: Reciprocal of sine, odd function

Trigonometric Equation: Multiple solutions due to periodicity, use identities | Algebraic Equation: Typically finite solutions, use algebraic manipulation

$r = f(\theta)$: Polar graph, represents distance from origin as a function of angle | $y = f(x)$: Cartesian graph, represents vertical position as a function of horizontal position

Transformations affect both functions similarly, but their initial starting points differ (sine starts at 0, cosine starts at 1).