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.

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).