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).
What is the formula for sine?
\(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
What is the formula for cosine?
\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
What is the formula for tangent?
\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
What is the formula for cosecant?
\(\csc(\theta) = \frac{1}{\sin(\theta)}\)
What is the formula for secant?
\(\sec(\theta) = \frac{1}{\cos(\theta)}\)
What is the formula for cotangent?
\(\cot(\theta) = \frac{1}{\tan(\theta)}\)
What is the conversion from polar to rectangular coordinates for x?
\(x = r \cos(\theta)\)
What is the conversion from polar to rectangular coordinates for y?
\(y = r \sin(\theta)\)
What is the Pythagorean identity?
\(\sin^2(\theta) + \cos^2(\theta) = 1\)
Formula for the period of \(y = A\sin(B(x-C)) + D\)?
\(\frac{2\pi}{|B|}\)
Explain the relationship between sine and cosine functions.
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.
Explain the effect of amplitude on a sinusoidal function.
Amplitude determines the maximum displacement of the function from its midline. A larger amplitude means a taller graph.
Explain the effect of the period on a sinusoidal function.
Period determines the length of one complete cycle. A smaller period means the function oscillates more rapidly.
Explain how transformations affect the graph of \(\sin(x)\).
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.
Explain why inverse trigonometric functions have restricted ranges.
To ensure they are functions (pass the vertical line test), their domains are restricted to principal values.
Explain how to convert from rectangular to polar coordinates.
Use the formulas \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan(\frac{y}{x})\), considering the quadrant of (x, y).
Explain the relationship between tangent and slope.
The tangent of an angle is equal to the slope of a line making that angle with the x-axis.
Explain the concept of phase shift in sinusoidal functions.
Phase shift is a horizontal translation of the sinusoidal function, affecting its starting point.
Explain the importance of the unit circle.
The unit circle provides a visual representation of trigonometric functions, allowing for easy determination of values at special angles.
Explain how polar coordinates are useful.
Polar coordinates are useful for representing circular or spiral paths and simplifying equations with circular symmetry.
