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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
The absolute value is the distance between a number and zero. The distance between and is .
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
The absolute value is the distance between a number and zero. The distance between and is .
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
Divide by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the right)
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
Simplify each term.
Step 6.1.2.1.1.1
Apply the distributive property.
Step 6.1.2.1.1.2
Cancel the common factor of .
Step 6.1.2.1.1.2.1
Factor out of .
Step 6.1.2.1.1.2.2
Factor out of .
Step 6.1.2.1.1.2.3
Cancel the common factor.
Step 6.1.2.1.1.2.4
Rewrite the expression.
Step 6.1.2.1.1.3
Combine and .
Step 6.1.2.1.1.4
Multiply by .
Step 6.1.2.1.2
Subtract from .
Step 6.1.2.1.3
Add and .
Step 6.1.2.1.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.
Step 6.1.2.1.5
The exact value of is .
Step 6.1.2.1.6
Multiply .
Step 6.1.2.1.6.1
Multiply by .
Step 6.1.2.1.6.2
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 right)
Vertical Shift:
Step 8