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Trigonometry Examples
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
Step 3.1
Find the period of .
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 .
Step 3.1.5.1
Factor out of .
Step 3.1.5.2
Cancel the common factor.
Step 3.1.5.3
Rewrite the expression.
Step 3.1.6
Multiply by .
Step 3.2
Find the period of .
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 .
Step 3.2.5.1
Factor out of .
Step 3.2.5.2
Cancel the common factor.
Step 3.2.5.3
Rewrite the expression.
Step 3.2.6
Multiply by .
Step 3.3
The period of addition/subtraction of trig functions is the maximum of the individual periods.
Step 4
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
Multiply .
Step 4.4.1
Combine and .
Phase Shift:
Step 4.4.2
Multiply by .
Phase Shift:
Phase Shift:
Step 4.5
Move the negative in front of the fraction.
Phase Shift:
Step 4.6
Replace with an approximation.
Phase Shift:
Step 4.7
Divide by .
Phase Shift:
Step 4.8
Multiply by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
Step 6
Step 6.1
Find the point at .
Step 6.1.1
Replace the variable with in the expression.
Step 6.1.2
Simplify the result.
Step 6.1.2.1
Simplify each term.
Step 6.1.2.1.1
Replace with decimal approximation.
Step 6.1.2.1.2
Multiply by .
Step 6.1.2.1.3
Divide by .
Step 6.1.2.1.4
Add and .
Step 6.1.2.1.5
Multiply by .
Step 6.1.2.2
Add and .
Step 6.1.2.3
The final answer is .
Step 6.2
List the points in a table.
Step 7
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
Step 8