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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
Cancel the common factor of .
Step 3.1.4.1
Cancel the common factor.
Step 3.1.4.2
Divide 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
Cancel the common factor of .
Step 3.2.4.1
Cancel the common factor.
Step 3.2.4.2
Divide 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
Move the negative in front of the fraction.
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
Cancel the common factor of .
Step 6.1.2.1.1.1
Move the leading negative in into the numerator.
Step 6.1.2.1.1.2
Cancel the common factor.
Step 6.1.2.1.1.3
Rewrite the expression.
Step 6.1.2.1.2
Add and .
Step 6.1.2.1.3
The exact value of is .
Step 6.1.2.1.4
Multiply by .
Step 6.1.2.2
Subtract from .
Step 6.1.2.3
The final answer is .
Step 6.2
Find the point at .
Step 6.2.1
Replace the variable with in the expression.
Step 6.2.2
Simplify the result.
Step 6.2.2.1
Simplify each term.
Step 6.2.2.1.1
Simplify each term.
Step 6.2.2.1.1.1
Apply the distributive property.
Step 6.2.2.1.1.2
Combine and .
Step 6.2.2.1.1.3
Cancel the common factor of .
Step 6.2.2.1.1.3.1
Move the leading negative in into the numerator.
Step 6.2.2.1.1.3.2
Cancel the common factor.
Step 6.2.2.1.1.3.3
Rewrite the expression.
Step 6.2.2.1.2
Add and .
Step 6.2.2.1.3
Add and .
Step 6.2.2.1.4
The exact value of is .
Step 6.2.2.1.5
Multiply by .
Step 6.2.2.2
Subtract from .
Step 6.2.2.3
The final answer is .
Step 6.3
Find the point at .
Step 6.3.1
Replace the variable with in the expression.
Step 6.3.2
Simplify the result.
Step 6.3.2.1
Simplify each term.
Step 6.3.2.1.1
Simplify each term.
Step 6.3.2.1.1.1
Apply the distributive property.
Step 6.3.2.1.1.2
Multiply by .
Step 6.3.2.1.1.3
Cancel the common factor of .
Step 6.3.2.1.1.3.1
Move the leading negative in into the numerator.
Step 6.3.2.1.1.3.2
Cancel the common factor.
Step 6.3.2.1.1.3.3
Rewrite the expression.
Step 6.3.2.1.2
Add and .
Step 6.3.2.1.3
Add and .
Step 6.3.2.1.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.3.2.1.5
The exact value of is .
Step 6.3.2.1.6
Multiply by .
Step 6.3.2.2
Subtract from .
Step 6.3.2.3
The final answer is .
Step 6.4
Find the point at .
Step 6.4.1
Replace the variable with in the expression.
Step 6.4.2
Simplify the result.
Step 6.4.2.1
Simplify each term.
Step 6.4.2.1.1
Simplify each term.
Step 6.4.2.1.1.1
Apply the distributive property.
Step 6.4.2.1.1.2
Combine and .
Step 6.4.2.1.1.3
Cancel the common factor of .
Step 6.4.2.1.1.3.1
Move the leading negative in into the numerator.
Step 6.4.2.1.1.3.2
Cancel the common factor.
Step 6.4.2.1.1.3.3
Rewrite the expression.
Step 6.4.2.1.1.4
Move to the left of .
Step 6.4.2.1.2
Add and .
Step 6.4.2.1.3
Add and .
Step 6.4.2.1.4
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 6.4.2.1.5
The exact value of is .
Step 6.4.2.1.6
Multiply .
Step 6.4.2.1.6.1
Multiply by .
Step 6.4.2.1.6.2
Multiply by .
Step 6.4.2.2
Subtract from .
Step 6.4.2.3
The final answer is .
Step 6.5
Find the point at .
Step 6.5.1
Replace the variable with in the expression.
Step 6.5.2
Simplify the result.
Step 6.5.2.1
Simplify each term.
Step 6.5.2.1.1
Simplify each term.
Step 6.5.2.1.1.1
Apply the distributive property.
Step 6.5.2.1.1.2
Move to the left of .
Step 6.5.2.1.1.3
Cancel the common factor of .
Step 6.5.2.1.1.3.1
Move the leading negative in into the numerator.
Step 6.5.2.1.1.3.2
Cancel the common factor.
Step 6.5.2.1.1.3.3
Rewrite the expression.
Step 6.5.2.1.2
Add and .
Step 6.5.2.1.3
Add and .
Step 6.5.2.1.4
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.1.5
The exact value of is .
Step 6.5.2.1.6
Multiply by .
Step 6.5.2.2
Subtract from .
Step 6.5.2.3
The final answer is .
Step 6.6
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