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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
Cancel the common factor of .
Step 3.4.1
Cancel the common factor.
Step 3.4.2
Divide by .
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:
Step 4.5
Move to the left of .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
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
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
Factor out of .
Step 6.1.2.1.1.3
Cancel the common factor.
Step 6.1.2.1.1.4
Rewrite the expression.
Step 6.1.2.1.2
Move the negative in front of the fraction.
Step 6.1.2.2
Combine fractions.
Step 6.1.2.2.1
Combine the numerators over the common denominator.
Step 6.1.2.2.2
Simplify the expression.
Step 6.1.2.2.2.1
Add and .
Step 6.1.2.2.2.2
Divide by .
Step 6.1.2.3
The exact value of is .
Step 6.1.2.4
Multiply by .
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
Apply the distributive property.
Step 6.2.2.1.2
Combine and .
Step 6.2.2.1.3
Cancel the common factor of .
Step 6.2.2.1.3.1
Move the leading negative in into the numerator.
Step 6.2.2.1.3.2
Factor out of .
Step 6.2.2.1.3.3
Cancel the common factor.
Step 6.2.2.1.3.4
Rewrite the expression.
Step 6.2.2.1.4
Move the negative in front of the fraction.
Step 6.2.2.2
Combine fractions.
Step 6.2.2.2.1
Combine the numerators over the common denominator.
Step 6.2.2.2.2
Simplify by adding numbers.
Step 6.2.2.2.2.1
Add and .
Step 6.2.2.2.2.2
Add and .
Step 6.2.2.3
The exact value of is .
Step 6.2.2.4
Multiply by .
Step 6.2.2.5
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
Apply the distributive property.
Step 6.3.2.1.2
Multiply by .
Step 6.3.2.1.3
Cancel the common factor of .
Step 6.3.2.1.3.1
Move the leading negative in into the numerator.
Step 6.3.2.1.3.2
Factor out of .
Step 6.3.2.1.3.3
Cancel the common factor.
Step 6.3.2.1.3.4
Rewrite the expression.
Step 6.3.2.1.4
Move the negative in front of the fraction.
Step 6.3.2.2
Combine fractions.
Step 6.3.2.2.1
Combine the numerators over the common denominator.
Step 6.3.2.2.2
Simplify the expression.
Step 6.3.2.2.2.1
Add and .
Step 6.3.2.2.2.2
Divide by .
Step 6.3.2.2.2.3
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 .
Step 6.3.2.5.1
Multiply by .
Step 6.3.2.5.2
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
Simplify each term.
Step 6.4.2.1.1
Apply the distributive property.
Step 6.4.2.1.2
Combine and .
Step 6.4.2.1.3
Cancel the common factor of .
Step 6.4.2.1.3.1
Move the leading negative in into the numerator.
Step 6.4.2.1.3.2
Factor out of .
Step 6.4.2.1.3.3
Cancel the common factor.
Step 6.4.2.1.3.4
Rewrite the expression.
Step 6.4.2.1.4
Simplify each term.
Step 6.4.2.1.4.1
Move to the left of .
Step 6.4.2.1.4.2
Move the negative in front of the fraction.
Step 6.4.2.2
Combine fractions.
Step 6.4.2.2.1
Combine the numerators over the common denominator.
Step 6.4.2.2.2
Simplify by adding numbers.
Step 6.4.2.2.2.1
Add and .
Step 6.4.2.2.2.2
Add and .
Step 6.4.2.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.4.2.4
The exact value of is .
Step 6.4.2.5
Multiply by .
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
Apply the distributive property.
Step 6.5.2.1.2
Move to the left of .
Step 6.5.2.1.3
Cancel the common factor of .
Step 6.5.2.1.3.1
Move the leading negative in into the numerator.
Step 6.5.2.1.3.2
Factor out of .
Step 6.5.2.1.3.3
Cancel the common factor.
Step 6.5.2.1.3.4
Rewrite the expression.
Step 6.5.2.1.4
Move the negative in front of the fraction.
Step 6.5.2.2
Combine fractions.
Step 6.5.2.2.1
Combine the numerators over the common denominator.
Step 6.5.2.2.2
Simplify the expression.
Step 6.5.2.2.2.1
Add and .
Step 6.5.2.2.2.2
Divide by .
Step 6.5.2.2.2.3
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
Multiply by .
Step 6.5.2.6
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: None
Step 8