Sine and Cosine Functions Quiz

Study Notes

Sine and Cosine Functions

Graphs of y = sin x and y = cos x: These functions are periodic, repeating their pattern every 2π radians (360 degrees). The graphs are wave-like.
Ranges of sin x and cos x: The range for both sin x and cos x is -1 ≤ y ≤ 1. This means the outputs of these functions are always between -1 and 1, inclusive.
Periods of sin x and cos x: The period for both functions is 2π.
Definitions of Amplitude: The amplitude of y = a sin x or y= a cos x is half the distance between the maximum and minimum values of the function; defined as |a|.

Parent Sine and Cosine Functions

Domain of Sin x and Cos x: Both functions have a domain of (-∞, ∞).
x-intercepts for sin x: occur at (nπ, 0) where n is an integer.
y-intercept for sin x: (0, 0)
Odd function for sin x: sin(-x) = -sin(x)
Origin symmetry for sin x
x-intercepts for cos x: occur at (nπ + π/2, 0) where n is an integer.
y-intercept for cos x: (0, 1)
Even function for cos x: cos(-x) = cos(x)
y-axis symmetry for cos x

Transformations of Sine and Cosine Functions

General form of sine and cosine transformations:

y = a sin(bx - c) + d y = a cos(bx - c) + d
Characteristics of sine and cosine functions (y = a sin(bx - c) + d or y= a cos(bx-c)+d):
- Amplitude = |a|
- Period = 2π/|b|
- Phase shift = c/b
- Vertical shift = d
Finding the left and right endpoints of a one-cycle interval: This is determined by solving bx - c = 0 and bx - c = 2π.
Horizontal translations: Shifting the graph left or right (phase shift) is determined by the 'c' value in the equation.
Vertical translations: Shifting the graph up or down is determined by the 'd' value.
Vertical shrinking and stretching: Determined by 'a' value in the equation.

Description

Test your understanding of sine and cosine functions, including their graphs, ranges, periods, and amplitudes. This quiz will cover the parent functions and their properties. Prepare to explore the periodic nature of these fundamental trigonometric functions!