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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 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
Divide 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 each term.
Step 6.1.2.1.1
Multiply by .
Step 6.1.2.1.2
Multiply by .
Step 6.1.2.2
Add and .
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
Multiply .
Step 6.2.2.1.1.1
Combine and .
Step 6.2.2.1.1.2
Raise to the power of .
Step 6.2.2.1.1.3
Raise to the power of .
Step 6.2.2.1.1.4
Use the power rule to combine exponents.
Step 6.2.2.1.1.5
Add and .
Step 6.2.2.1.2
Cancel the common factor of .
Step 6.2.2.1.2.1
Factor out of .
Step 6.2.2.1.2.2
Cancel the common factor.
Step 6.2.2.1.2.3
Rewrite the expression.
Step 6.2.2.1.3
Rewrite as .
Step 6.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.2.2.3.1
Multiply by .
Step 6.2.2.3.2
Reorder the factors of .
Step 6.2.2.4
Combine the numerators over the common denominator.
Step 6.2.2.5
Multiply by .
Step 6.2.2.6
Evaluate .
Step 6.2.2.7
Multiply by .
Step 6.2.2.8
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
Multiply .
Step 6.3.2.1.1.1
Combine and .
Step 6.3.2.1.1.2
Raise to the power of .
Step 6.3.2.1.1.3
Raise to the power of .
Step 6.3.2.1.1.4
Use the power rule to combine exponents.
Step 6.3.2.1.1.5
Add and .
Step 6.3.2.1.2
Combine and .
Step 6.3.2.1.3
Move the negative in front of the fraction.
Step 6.3.2.2
Combine the numerators over the common denominator.
Step 6.3.2.3
Apply the reference angle by finding the angle with equivalent trig values in the first 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
Simplify each term.
Step 6.4.2.1.1
Multiply .
Step 6.4.2.1.1.1
Combine and .
Step 6.4.2.1.1.2
Raise to the power of .
Step 6.4.2.1.1.3
Raise to the power of .
Step 6.4.2.1.1.4
Use the power rule to combine exponents.
Step 6.4.2.1.1.5
Add and .
Step 6.4.2.1.2
Cancel the common factor of .
Step 6.4.2.1.2.1
Factor out of .
Step 6.4.2.1.2.2
Cancel the common factor.
Step 6.4.2.1.2.3
Rewrite the expression.
Step 6.4.2.1.3
Rewrite as .
Step 6.4.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.4.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.4.2.3.1
Multiply by .
Step 6.4.2.3.2
Reorder the factors of .
Step 6.4.2.4
Combine the numerators over the common denominator.
Step 6.4.2.5
Multiply by .
Step 6.4.2.6
Evaluate .
Step 6.4.2.7
Multiply by .
Step 6.4.2.8
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
Multiply .
Step 6.5.2.1.1.1
Combine and .
Step 6.5.2.1.1.2
Raise to the power of .
Step 6.5.2.1.1.3
Raise to the power of .
Step 6.5.2.1.1.4
Use the power rule to combine exponents.
Step 6.5.2.1.1.5
Add and .
Step 6.5.2.1.2
Multiply .
Step 6.5.2.1.2.1
Combine and .
Step 6.5.2.1.2.2
Multiply by .
Step 6.5.2.1.3
Move the negative in front of the fraction.
Step 6.5.2.2
Combine the numerators over the common denominator.
Step 6.5.2.3
Evaluate .
Step 6.5.2.4
Multiply by .
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: None
Vertical Shift: None
Step 8