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What does the graph of \(y = \sin(x)\) tell us?
It shows a periodic wave oscillating between -1 and 1, with x-intercepts at multiples of \(\pi\).
How does the graph of \(y = \cos(x)\) differ from \(y = \sin(x)\)?
The graph of cosine is a sine wave shifted \(\frac{\pi}{2}\) units to the left. It starts at its maximum value.
What does a vertical stretch in the graph of \(y = \sin(x)\) indicate?
It indicates a change in amplitude, making the wave taller or shorter.
What does the graph of \(r = \cos(\theta)\) represent in polar coordinates?
It represents a circle centered on the x-axis passing through the origin.
What does the graph of \(y = \tan(x)\) tell us?
It has vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\), where n is an integer, and a period of \(\pi\).
How does the graph of \(y = \arcsin(x)\) look?
It's the inverse of the sine function, defined for [-1, 1] and ranging from \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
What does the graph of \(r = a\) in polar coordinates represent?
A circle centered at the origin with radius a.
What does a horizontal compression of \(\sin(x)\) look like on a graph?
The period decreases, causing the graph to oscillate more frequently.
How can you identify the period of a sinusoidal function from its graph?
Measure the distance between two consecutive peaks or troughs.
What does the graph of \(r = \theta\) in polar coordinates look like?
A spiral that extends outward from the origin as \(\theta\) increases.
Define periodic phenomena.
Patterns that repeat over time.
What are sinusoidal functions?
Periodic functions represented by sine or cosine.
Define inverse trigonometric functions.
Functions that find angles from given sine, cosine, or tangent values.
What are polar coordinates?
Another way to express positions in a plane using distance and angle.
Define amplitude of a sinusoidal function.
The maximum displacement from the midline of the function.
What is the period of a trigonometric function?
The length of one complete cycle of the function.
Define vertical shift.
A transformation that moves the graph of a function up or down.
What is the unit circle?
A circle with radius 1 centered at the origin, used to define trigonometric functions.
Define radian.
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.
What is a polar function?
A function defined in polar coordinates, typically in the form r = f(θ).
How to find the amplitude and period of \(y = A\sin(Bx + C) + D\)?
Amplitude: |A|, Period: \(\frac{2\pi}{|B|}\)
How to solve \(\sin(x) = a\) for x?
Find the principal value using \(x = \arcsin(a)\), then use the properties of sine to find other solutions within the desired interval.
How to convert the point (3,4) from rectangular to polar coordinates?
1. Calculate r: \(r = \sqrt{3^2 + 4^2} = 5\). 2. Calculate \(\theta\): \(\theta = \arctan(\frac{4}{3}) \approx 0.93\) radians.
How to find the x-intercepts of \(y = \cos(2x)\)?
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.
How do you graph a polar equation of the form \(r = a\cos(\theta)\)?
Create a table of values for \(\theta\) and r, plot the points (r, \(\theta\)), and connect them to form the graph (usually a circle).
How to solve a trigonometric equation involving multiple angles?
Use trigonometric identities to simplify the equation, then solve for the multiple angle, and finally solve for the variable.
How to determine the vertical shift of a sinusoidal function?
Identify the midline of the function. The vertical shift is the distance between the midline and the x-axis.
How to find the period of a transformed tangent function?
For \(y = A\tan(Bx + C) + D\), the period is \(\frac{\pi}{|B|}\).
How to find the domain of an inverse trigonometric function?
Consider the range of the original trigonometric function. For example, the domain of \(\arcsin(x)\) is [-1, 1].
How do you convert rectangular equation to polar equation?
Replace x with \(r\cos(\theta)\) and y with \(r\sin(\theta)\), then simplify the equation.