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What are the differences between sine and cosine?
Sine: Starts at 0, odd function | Cosine: Starts at 1, even function
What are the differences between polar and rectangular coordinates?
Polar: Uses radius and angle, good for circles | Rectangular: Uses x and y, good for lines
What are the differences between \(\sin^{-1}(x)\) and \(\frac{1}{\sin(x)}\)?
\(\sin^{-1}(x)\): Inverse sine, returns an angle | \(\frac{1}{\sin(x)}\): Cosecant, reciprocal of sine
What are the differences between amplitude and period?
Amplitude: Vertical distance from midline to max/min | Period: Horizontal length of one cycle
What are the differences between phase shift and vertical shift?
Phase Shift: Horizontal translation | Vertical Shift: Vertical translation
What are the differences between tangent and cotangent?
Tangent: \(\frac{\sin(x)}{\cos(x)}\), asymptotes at \(\frac{\pi}{2} + n\pi\) | Cotangent: \(\frac{\cos(x)}{\sin(x)}\), asymptotes at \(n\pi\)
What are the differences between secant and cosecant?
Secant: Reciprocal of cosine, even function | Cosecant: Reciprocal of sine, odd function
What are the differences between solving a trigonometric equation and an algebraic equation?
Trigonometric Equation: Multiple solutions due to periodicity, use identities | Algebraic Equation: Typically finite solutions, use algebraic manipulation
What are the differences between \(r = f(\theta)\) and \(y = f(x)\) in terms of graphs?
\(r = f(\theta)\): 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
What are the differences between transformations of sine and cosine functions?
Transformations affect both functions similarly, but their initial starting points differ (sine starts at 0, cosine starts at 1).
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.
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(ฮธ).