Trigonometry Examples

Find Amplitude, Period, and Phase Shift y=5sin((2pi)/365*(x-77))+12
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
Multiply the numerator by the reciprocal of the denominator.
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
Multiply the numerator by the reciprocal of the denominator.
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
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Step 4.4
Cancel the common factor of .
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Step 4.4.1
Factor out of .
Phase Shift:
Step 4.4.2
Cancel the common factor.
Phase Shift:
Step 4.4.3
Rewrite the expression.
Phase Shift:
Phase Shift:
Step 4.5
Cancel the common factor of .
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Step 4.5.1
Cancel the common factor.
Phase Shift:
Step 4.5.2
Rewrite the expression.
Phase Shift:
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the right)
Vertical Shift:
Step 6