How would I state the phase shift of y=cos(θ-pi/3) & then graph it?
Step by step would be awesome, I was out sick and needed to do a worksheet to catch up!
first make a sketch of y=cosθ
then translate it pi/3 radians to the right
in general for y=cos(θ-k) move y=cos(θ)to the right k radians
or for y=cos(θ+k) move y=cos(θ)to the left k radians
if there is a coefficient in front of the angle you have to factor out that coefficient to show the phase shift
e.g. y=cos(3θ-pi/2)
becomes y=cos3(θ-pi/6) , so the phase shift is pi/5 to the right
last line should have said:
becomes y=cos3(θ-pi/6) , so the phase shift is pi/6 to the right
(keystroke error)
To determine the phase shift of the function y = cos(θ - π/3), we need to understand the general equation for the phase shift of a cosine function, which is given by y = cos(θ - φ), where φ represents the phase shift.
Step 1: Identify the value of φ in the equation y = cos(θ - π/3).
In this case, φ is equal to π/3.
Step 2: Interpret the value of φ.
Since φ is positive, it indicates a shift to the right. In other words, the graph of the function y = cos(θ - π/3) is shifted horizontally to the right by an amount of π/3 units.
Now, let's proceed to graphing the function:
Step 1: Determine the amplitude.
Since there is no coefficient preceding the cosine function, the amplitude is 1.
Step 2: Sketch the unshifted cosine function.
To graph the unshifted cosine function, plot the key points for one full period: 0°, 90°, 180°, 270°, and 360°. The corresponding y-values are 1, 0, -1, 0, and 1, respectively.
Step 3: Apply the phase shift.
Shift the graph horizontally π/3 units to the right. To do this, add π/3 to each of the x-values obtained in Step 2.
Step 4: Sketch the shifted graph.
Plot the new points obtained in Step 3 and connect them with a smooth curve.
Your final graph represents the function y = cos(θ - π/3) with a phase shift to the right by π/3 units.