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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
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.1.4
Cancel the common factor of and .
Step 3.1.4.1
Factor out of .
Step 3.1.4.2
Cancel the common factors.
Step 3.1.4.2.1
Factor out of .
Step 3.1.4.2.2
Cancel the common factor.
Step 3.1.4.2.3
Rewrite the expression.
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.4
Cancel the common factor of and .
Step 3.2.4.1
Factor out of .
Step 3.2.4.2
Cancel the common factors.
Step 3.2.4.2.1
Factor out of .
Step 3.2.4.2.2
Cancel the common factor.
Step 3.2.4.2.3
Rewrite the expression.
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
Cancel the common factor of and .
Step 4.3.1
Factor out of .
Phase Shift:
Step 4.3.2
Cancel the common factors.
Step 4.3.2.1
Factor out of .
Phase Shift:
Step 4.3.2.2
Cancel the common factor.
Phase Shift:
Step 4.3.2.3
Rewrite the expression.
Phase Shift:
Phase Shift:
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the right)
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
Simplify each term.
Step 6.1.2.1.1.1
Cancel the common factor of .
Step 6.1.2.1.1.1.1
Factor out of .
Step 6.1.2.1.1.1.2
Cancel the common factor.
Step 6.1.2.1.1.1.3
Rewrite the expression.
Step 6.1.2.1.1.2
Multiply by .
Step 6.1.2.1.2
Subtract from .
Step 6.1.2.1.3
The exact value of is .
Step 6.1.2.1.3.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 6.1.2.1.3.2
Apply the sine half-angle identity.
Step 6.1.2.1.3.3
Change the to because sine is positive in the first quadrant.
Step 6.1.2.1.3.4
Simplify .
Step 6.1.2.1.3.4.1
The exact value of is .
Step 6.1.2.1.3.4.2
Multiply by .
Step 6.1.2.1.3.4.3
Subtract from .
Step 6.1.2.1.3.4.4
Divide by .
Step 6.1.2.1.3.4.5
Rewrite as .
Step 6.1.2.1.3.4.6
Pull terms out from under the radical, assuming positive real numbers.
Step 6.1.2.1.4
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
Simplify each term.
Step 6.2.2.1.1.1
Apply the distributive property.
Step 6.2.2.1.1.2
Cancel the common factor of .
Step 6.2.2.1.1.2.1
Factor out of .
Step 6.2.2.1.1.2.2
Cancel the common factor.
Step 6.2.2.1.1.2.3
Rewrite the expression.
Step 6.2.2.1.1.3
Cancel the common factor of .
Step 6.2.2.1.1.3.1
Factor out of .
Step 6.2.2.1.1.3.2
Cancel the common factor.
Step 6.2.2.1.1.3.3
Rewrite the expression.
Step 6.2.2.1.1.4
Multiply by .
Step 6.2.2.1.2
Subtract from .
Step 6.2.2.1.3
Add and .
Step 6.2.2.2
The final answer is .
Step 6.2.3
Convert to a decimal.
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
Simplify each term.
Step 6.3.2.1.1.1
Apply the distributive property.
Step 6.3.2.1.1.2
Cancel the common factor of .
Step 6.3.2.1.1.2.1
Cancel the common factor.
Step 6.3.2.1.1.2.2
Rewrite the expression.
Step 6.3.2.1.1.3
Cancel the common factor of .
Step 6.3.2.1.1.3.1
Factor out of .
Step 6.3.2.1.1.3.2
Cancel the common factor.
Step 6.3.2.1.1.3.3
Rewrite the expression.
Step 6.3.2.1.1.4
Multiply by .
Step 6.3.2.1.2
Replace with decimal approximation.
Step 6.3.2.1.3
Subtract from .
Step 6.3.2.1.4
Add and .
Step 6.3.2.2
The final answer is .
Step 6.3.3
Convert to a decimal.
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
Simplify each term.
Step 6.4.2.1.1.1
Apply the distributive property.
Step 6.4.2.1.1.2
Cancel the common factor of .
Step 6.4.2.1.1.2.1
Factor out of .
Step 6.4.2.1.1.2.2
Cancel the common factor.
Step 6.4.2.1.1.2.3
Rewrite the expression.
Step 6.4.2.1.1.3
Cancel the common factor of .
Step 6.4.2.1.1.3.1
Factor out of .
Step 6.4.2.1.1.3.2
Cancel the common factor.
Step 6.4.2.1.1.3.3
Rewrite the expression.
Step 6.4.2.1.1.4
Multiply by .
Step 6.4.2.1.2
Subtract from .
Step 6.4.2.1.3
Add and .
Step 6.4.2.2
The final answer is .
Step 6.4.3
Convert to a decimal.
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
Simplify each term.
Step 6.5.2.1.1.1
Apply the distributive property.
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 6.5.2.1.1.3
Cancel the common factor of .
Step 6.5.2.1.1.3.1
Factor out of .
Step 6.5.2.1.1.3.2
Cancel the common factor.
Step 6.5.2.1.1.3.3
Rewrite the expression.
Step 6.5.2.1.1.4
Multiply by .
Step 6.5.2.1.2
Subtract from .
Step 6.5.2.1.3
Add and .
Step 6.5.2.2
The final answer is .
Step 6.5.3
Convert to a decimal.
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:
Step 8