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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
Simplify the denominator.
Step 3.3.1
is approximately which is positive so remove the absolute value
Step 3.3.2
Write as a fraction with a common denominator.
Step 3.3.3
Combine the numerators over the common denominator.
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
Combine and .
Step 3.5.3
Raise to the power of .
Step 3.5.4
Raise to the power of .
Step 3.5.5
Use the power rule to combine exponents.
Step 3.5.6
Add 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
Simplify the denominator.
Step 4.3.1
Write as a fraction with a common denominator.
Phase Shift:
Step 4.3.2
Combine the numerators over the common denominator.
Phase Shift:
Phase Shift:
Step 4.4
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Step 4.5
Multiply by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: None
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 the numerator.
Step 6.1.2.1.1
Simplify each term.
Step 6.1.2.1.1.1
Divide by .
Step 6.1.2.1.1.2
Multiply by .
Step 6.1.2.1.2
Add and .
Step 6.1.2.1.3
The exact value of is .
Step 6.1.2.2
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 the numerator.
Step 6.2.2.1.1
Simplify each term.
Step 6.2.2.1.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.2.2.1.1.2
Combine.
Step 6.2.2.1.1.3
Cancel the common factor of and .
Step 6.2.2.1.1.3.1
Factor out of .
Step 6.2.2.1.1.3.2
Cancel the common factors.
Step 6.2.2.1.1.3.2.1
Factor out of .
Step 6.2.2.1.1.3.2.2
Cancel the common factor.
Step 6.2.2.1.1.3.2.3
Rewrite the expression.
Step 6.2.2.1.1.4
Multiply by .
Step 6.2.2.1.2
Evaluate .
Step 6.2.2.2
Divide by .
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 the numerator.
Step 6.3.2.1.1
Simplify each term.
Step 6.3.2.1.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.2.1.1.2
Cancel the common factor of .
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 6.3.2.1.2
Combine the numerators over the common denominator.
Step 6.3.2.1.3
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 6.3.2.1.4
The exact value of is .
Step 6.3.2.1.5
Multiply by .
Step 6.3.2.2
Move the negative in front of the fraction.
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 the numerator.
Step 6.4.2.1.1
Simplify each term.
Step 6.4.2.1.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.4.2.1.1.2
Combine.
Step 6.4.2.1.1.3
Cancel the common factor of and .
Step 6.4.2.1.1.3.1
Factor out of .
Step 6.4.2.1.1.3.2
Cancel the common factors.
Step 6.4.2.1.1.3.2.1
Factor out of .
Step 6.4.2.1.1.3.2.2
Cancel the common factor.
Step 6.4.2.1.1.3.2.3
Rewrite the expression.
Step 6.4.2.1.1.4
Multiply by .
Step 6.4.2.1.2
Evaluate .
Step 6.4.2.2
Divide by .
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 the numerator.
Step 6.5.2.1.1
Simplify each term.
Step 6.5.2.1.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.5.2.1.1.2
Cancel the common factor of .
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 6.5.2.1.2
Combine the numerators over the common denominator.
Step 6.5.2.1.3
Evaluate .
Step 6.5.2.2
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: None
Step 8