Skip to main content
Back

Graphs and Properties of Sine and Cosine Functions

Control buttons has been changed to "navigation" mode.
1/20

Terms in this set (20)

  • Domain of sine and cosine functions
    The domain of both sine and cosine functions is the set of all real numbers.
  • Range of sine and cosine functions
    The range of sine and cosine functions is all real numbers from −1 to 1, inclusive.
  • Period of sine and cosine functions
    Both sine and cosine functions have a period of \(2\pi\).
  • Symmetry of sine function
    The sine function is an odd function, symmetric with respect to the origin.
  • Symmetry of cosine function
    The cosine function is an even function, symmetric with respect to the y-axis.
  • Maximum and minimum values of sine and cosine
    Both sine and cosine functions have a maximum value of 1 and a minimum value of −1.
  • Key x-intercepts of sine function
    The sine function has x-intercepts at \(0, \pi, 2\pi\) and multiples thereof.
  • Key x-intercepts of cosine function
    The cosine function has x-intercepts at \(\frac{\pi}{2}, \frac{3\pi}{2}\) and odd multiples thereof.
  • Effect of amplitude on sine and cosine graphs
    Amplitude A stretches the graph vertically by a factor of |A|, changing the range to [−|A|, |A|].
  • Effect of frequency (omega) on period
    For functions of the form y = A sin(ωx) or y = A cos(ωx), the period is \(\frac{2\pi}{\omega}\).
  • Graphing sine function using key points
    Divide one period into four equal subintervals to find five key points for plotting the sine curve.
  • Graphing cosine function using key points
    Divide one period into four equal subintervals to find five key points for plotting the cosine curve.
  • Vertical shift in sinusoidal functions
    Adding a constant D to y = A sin(ωx) + D or y = A cos(ωx) + D shifts the graph vertically by D units.
  • Horizontal compression/stretch in sinusoidal functions
    Multiplying the input by ω compresses the graph horizontally by a factor of 1/ω.
  • Equation form of sinusoidal functions
    General form: y = A sin(ωx + φ) + D or y = A cos(ωx + φ) + D, where φ is phase shift.
  • Amplitude from sinusoidal equation
    The amplitude is the absolute value of A in the sinusoidal function.
  • Period from sinusoidal equation
    The period is calculated as \(\frac{2\pi}{|\omega|}\).
  • Finding sinusoidal equation from graph
    Identify amplitude, period, phase shift, and vertical shift from the graph to write the equation.
  • Sine function key points in one period
    At \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\), sine values are 0, 1, 0, −1, 0 respectively.
  • Cosine function key points in one period
    At \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\), cosine values are 1, 0, −1, 0, 1 respectively.