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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
Multiply .
Step 3.1.5.1
Raise to the power of .
Step 3.1.5.2
Raise to the power of .
Step 3.1.5.3
Use the power rule to combine exponents.
Step 3.1.5.4
Add and .
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
Multiply .
Step 3.2.5.1
Raise to the power of .
Step 3.2.5.2
Raise to the power of .
Step 3.2.5.3
Use the power rule to combine exponents.
Step 3.2.5.4
Add and .
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 by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: None
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
Divide by .
Step 6.1.2.1.2
The exact value of is .
Step 6.1.2.1.3
Multiply by .
Step 6.1.2.2
Add and .
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
Multiply the numerator by the reciprocal of the denominator.
Step 6.2.2.1.2
Combine.
Step 6.2.2.1.3
Cancel the common factor of and .
Step 6.2.2.1.3.1
Factor out of .
Step 6.2.2.1.3.2
Cancel the common factors.
Step 6.2.2.1.3.2.1
Factor out of .
Step 6.2.2.1.3.2.2
Cancel the common factor.
Step 6.2.2.1.3.2.3
Rewrite the expression.
Step 6.2.2.1.4
Multiply by .
Step 6.2.2.1.5
The exact value of is .
Step 6.2.2.1.6
Multiply by .
Step 6.2.2.2
Add and .
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
Cancel the common factor of and .
Step 6.3.2.1.1.1
Factor out of .
Step 6.3.2.1.1.2
Cancel the common factors.
Step 6.3.2.1.1.2.1
Multiply by .
Step 6.3.2.1.1.2.2
Cancel the common factor.
Step 6.3.2.1.1.2.3
Rewrite the expression.
Step 6.3.2.1.1.2.4
Divide by .
Step 6.3.2.1.2
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.3.2.1.3
The exact value of is .
Step 6.3.2.1.4
Multiply by .
Step 6.3.2.2
Add and .
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
Multiply the numerator by the reciprocal of the denominator.
Step 6.4.2.1.2
Combine.
Step 6.4.2.1.3
Cancel the common factor of and .
Step 6.4.2.1.3.1
Factor out of .
Step 6.4.2.1.3.2
Cancel the common factors.
Step 6.4.2.1.3.2.1
Factor out of .
Step 6.4.2.1.3.2.2
Cancel the common factor.
Step 6.4.2.1.3.2.3
Rewrite the expression.
Step 6.4.2.1.4
Multiply by .
Step 6.4.2.1.5
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 6.4.2.1.6
The exact value of is .
Step 6.4.2.1.7
Multiply .
Step 6.4.2.1.7.1
Multiply by .
Step 6.4.2.1.7.2
Multiply by .
Step 6.4.2.2
Add and .
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
Cancel the common factor of and .
Step 6.5.2.1.1.1
Factor out of .
Step 6.5.2.1.1.2
Cancel the common factors.
Step 6.5.2.1.1.2.1
Multiply by .
Step 6.5.2.1.1.2.2
Cancel the common factor.
Step 6.5.2.1.1.2.3
Rewrite the expression.
Step 6.5.2.1.1.2.4
Divide by .
Step 6.5.2.1.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.1.3
The exact value of is .
Step 6.5.2.1.4
Multiply by .
Step 6.5.2.2
Add and .
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: None
Vertical Shift:
Step 8