Phase Shifts of Sine and Cosine Functions

Phase Shifts of Sine & Cosine

Introduction to Phase Shifts

Phase shifts are horizontal translations of trigonometric graphs, specifically the sine and cosine functions. Understanding phase shifts is essential for graphing and analyzing periodic functions in trigonometry.

• Phase shift refers to the horizontal movement of the graph, indicated by the value inside the parentheses of the function's argument.

• A phase shift can move a sine or cosine graph left or right along the x-axis.

General Form of Sine and Cosine Functions

The general form for sine and cosine functions with phase shift is:

• $y = A \sin(Bx - C) + D$

• $y = A \cos(Bx - C) + D$

Where:

• A: Amplitude (vertical stretch/shrink)

• B: Affects the period ($\text{Period} = \frac{2\pi}{B}$)

• C: Phase shift ($\text{Phase Shift} = \frac{C}{B}$)

• D: Vertical shift (up/down)

Phase Shift Calculation

To determine the phase shift:

• Set the inside of the parentheses equal to zero: $Bx - C = 0$

• Solve for $x$: $x = \frac{C}{B}$

• If $C > 0$, the graph shifts to the right; if $C < 0$, the graph shifts to the left.

Table: Phase Shift of Sine & Cosine Graphs

Function	Phase Shift	Direction
$y = \sin(x - c)$	$c$	Right
$y = \sin(x + c)$	$-c$	Left
$y = \cos(x - c)$	$c$	Right
$y = \cos(x + c)$	$-c$	Left

Graphical Example

Consider the function $y = \sin(x + 2)$. The phase shift is $-2$, so the graph shifts 2 units to the left.

• Amplitude: 1

• Period: $2\pi$

• Phase Shift: $-2$ (left)

• Vertical Shift: 0

Example: Graph $y = \sin(x + 2)$ and indicate the phase shift.

Practice Problems

• Describe the phase shift for $y = \cos(5x - \frac{\pi}{2})$.

  • Set $5x - \frac{\pi}{2} = 0$

  • $x = \frac{\pi}{10}$

  • Phase shift: $\frac{\pi}{10}$ units to the right

• Describe the phase shift for $y = \cos(2x + \frac{\pi}{3})$.

  • Set $2x + \frac{\pi}{3} = 0$

  • $x = -\frac{\pi}{6}$

  • Phase shift: $\frac{\pi}{6}$ units to the left

Summary Table: Key Properties of Sine & Cosine with Phase Shift

Property	Effect
Amplitude ($A$)	Vertical stretch/shrink
Period ($B$)	$\frac{2\pi}{B}$
Phase Shift ($C$)	$\frac{C}{B}$ units left/right
Vertical Shift ($D$)	Up/down movement

Additional info: The notes also include graphical examples and practice problems to reinforce the concept of phase shifts in trigonometric functions.