Trigonometry Examples

Find Amplitude, Period, and Phase Shift f(x)=7cos(4pix)+3
Step 1
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Step 2
Find the amplitude .
Amplitude:
Step 3
Find the period using the formula .
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Step 3.1
Find the period of .
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Step 3.1.1
The period of the function can be calculated using .
Step 3.1.2
Replace with in the formula for period.
Step 3.1.3
is approximately which is positive so remove the absolute value
Step 3.1.4
Cancel the common factor of and .
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Step 3.1.4.1
Factor out of .
Step 3.1.4.2
Cancel the common factors.
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Step 3.1.4.2.1
Factor out of .
Step 3.1.4.2.2
Cancel the common factor.
Step 3.1.4.2.3
Rewrite the expression.
Step 3.1.5
Cancel the common factor of .
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Step 3.1.5.1
Cancel the common factor.
Step 3.1.5.2
Rewrite the expression.
Step 3.2
Find the period of .
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Step 3.2.1
The period of the function can be calculated using .
Step 3.2.2
Replace with in the formula for period.
Step 3.2.3
is approximately which is positive so remove the absolute value
Step 3.2.4
Cancel the common factor of and .
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Step 3.2.4.1
Factor out of .
Step 3.2.4.2
Cancel the common factors.
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Step 3.2.4.2.1
Factor out of .
Step 3.2.4.2.2
Cancel the common factor.
Step 3.2.4.2.3
Rewrite the expression.
Step 3.2.5
Cancel the common factor of .
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Step 3.2.5.1
Cancel the common factor.
Step 3.2.5.2
Rewrite the expression.
Step 3.3
The period of addition/subtraction of trig functions is the maximum of the individual periods.
Step 4
Find the phase shift using the formula .
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Step 4.1
The phase shift of the function can be calculated from .
Phase Shift:
Step 4.2
Replace the values of and in the equation for phase shift.
Phase Shift:
Step 4.3
Cancel the common factor of and .
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Step 4.3.1
Factor out of .
Phase Shift:
Step 4.3.2
Cancel the common factors.
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Step 4.3.2.1
Factor out of .
Phase Shift:
Step 4.3.2.2
Cancel the common factor.
Phase Shift:
Step 4.3.2.3
Rewrite the expression.
Phase Shift:
Phase Shift:
Phase Shift:
Step 4.4
Divide by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: None
Vertical Shift:
Step 6