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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
The period of the function can be calculated using .
Step 3.2
Replace with in the formula for period.
Step 3.3
is approximately which is positive so remove the absolute value
Step 3.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.5
Multiply .
Step 3.5.1
Combine and .
Step 3.5.2
Multiply by .
Step 3.5.3
Combine and .
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
Cancel the common factor of .
Step 4.4.1
Cancel the common factor.
Phase Shift:
Step 4.4.2
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: None
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
Multiply by .
Step 6.1.2.2
Divide by .
Step 6.1.2.3
Subtract from .
Step 6.1.2.4
The exact value of is .
Step 6.1.2.5
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 the numerator.
Step 6.2.2.1.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.1.1.2
Combine and .
Step 6.2.2.1.1.3
Combine the numerators over the common denominator.
Step 6.2.2.1.1.4
Multiply by .
Step 6.2.2.1.2
Combine and .
Step 6.2.2.1.3
Reduce the expression by cancelling the common factors.
Step 6.2.2.1.3.1
Reduce the expression by cancelling the common factors.
Step 6.2.2.1.3.1.1
Cancel the common factor.
Step 6.2.2.1.3.1.2
Rewrite the expression.
Step 6.2.2.1.3.2
Divide by .
Step 6.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.3
Combine fractions.
Step 6.2.2.3.1
Combine and .
Step 6.2.2.3.2
Combine the numerators over the common denominator.
Step 6.2.2.4
Simplify the numerator.
Step 6.2.2.4.1
Multiply by .
Step 6.2.2.4.2
Subtract from .
Step 6.2.2.4.3
Add and .
Step 6.2.2.5
The exact value of is .
Step 6.2.2.6
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
Factor out of .
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 distributive property.
Step 6.3.2.1.3
Cancel the common factor of .
Step 6.3.2.1.3.1
Cancel the common factor.
Step 6.3.2.1.3.2
Rewrite the expression.
Step 6.3.2.1.4
Multiply by .
Step 6.3.2.2
Simplify by subtracting numbers.
Step 6.3.2.2.1
Subtract from .
Step 6.3.2.2.2
Add and .
Step 6.3.2.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 6.3.2.4
The exact value of is .
Step 6.3.2.5
Multiply by .
Step 6.3.2.6
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
To write as a fraction with a common denominator, multiply by .
Step 6.4.2.2
Combine fractions.
Step 6.4.2.2.1
Combine and .
Step 6.4.2.2.2
Combine the numerators over the common denominator.
Step 6.4.2.3
Simplify the numerator.
Step 6.4.2.3.1
Apply the distributive property.
Step 6.4.2.3.2
Multiply by .
Step 6.4.2.3.3
Multiply by .
Step 6.4.2.3.4
Subtract from .
Step 6.4.2.3.5
Add and .
Step 6.4.2.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.4.2.5
The exact value of is .
Step 6.4.2.6
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
Factor out of .
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
Apply the distributive property.
Step 6.5.2.1.3
Cancel the common factor of .
Step 6.5.2.1.3.1
Cancel the common factor.
Step 6.5.2.1.3.2
Rewrite the expression.
Step 6.5.2.1.4
Multiply by .
Step 6.5.2.2
Simplify by subtracting numbers.
Step 6.5.2.2.1
Subtract from .
Step 6.5.2.2.2
Add and .
Step 6.5.2.3
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.4
The exact value of is .
Step 6.5.2.5
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 right)
Vertical Shift: None
Step 8