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Trigonometry Examples
Step 1
Rewrite the expression as .
Step 2
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Step 3
Find the amplitude .
Amplitude:
Step 4
Step 4.1
Find the period of .
Step 4.1.1
The period of the function can be calculated using .
Step 4.1.2
Replace with in the formula for period.
Step 4.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.1.4
Divide by .
Step 4.2
Find the period of .
Step 4.2.1
The period of the function can be calculated using .
Step 4.2.2
Replace with in the formula for period.
Step 4.2.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.4
Divide by .
Step 4.3
The period of addition/subtraction of trig functions is the maximum of the individual periods.
Step 5
Step 5.1
The phase shift of the function can be calculated from .
Phase Shift:
Step 5.2
Replace the values of and in the equation for phase shift.
Phase Shift:
Step 5.3
Divide by .
Phase Shift:
Phase Shift:
Step 6
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the right)
Vertical Shift:
Step 7
Step 7.1
Find the point at .
Step 7.1.1
Replace the variable with in the expression.
Step 7.1.2
Simplify the result.
Step 7.1.2.1
Simplify each term.
Step 7.1.2.1.1
Subtract from .
Step 7.1.2.1.2
The exact value of is .
Step 7.1.2.1.2.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 7.1.2.1.2.2
Apply the cosine half-angle identity .
Step 7.1.2.1.2.3
Change the to because cosine is positive in the first quadrant.
Step 7.1.2.1.2.4
The exact value of is .
Step 7.1.2.1.2.5
Simplify .
Step 7.1.2.1.2.5.1
Add and .
Step 7.1.2.1.2.5.2
Divide by .
Step 7.1.2.1.2.5.3
Any root of is .
Step 7.1.2.1.3
Multiply by .
Step 7.1.2.2
Add and .
Step 7.1.2.3
The final answer is .
Step 7.2
Find the point at .
Step 7.2.1
Replace the variable with in the expression.
Step 7.2.2
Simplify the result.
Step 7.2.2.1
Subtract from .
Step 7.2.2.2
The final answer is .
Step 7.2.3
Convert to a decimal.
Step 7.3
Find the point at .
Step 7.3.1
Replace the variable with in the expression.
Step 7.3.2
Simplify the result.
Step 7.3.2.1
Subtract from .
Step 7.3.2.2
The final answer is .
Step 7.3.3
Convert to a decimal.
Step 7.4
Find the point at .
Step 7.4.1
Replace the variable with in the expression.
Step 7.4.2
Simplify the result.
Step 7.4.2.1
Subtract from .
Step 7.4.2.2
The final answer is .
Step 7.4.3
Convert to a decimal.
Step 7.5
Find the point at .
Step 7.5.1
Replace the variable with in the expression.
Step 7.5.2
Simplify the result.
Step 7.5.2.1
Subtract from .
Step 7.5.2.2
The final answer is .
Step 7.5.3
Convert to a decimal.
Step 7.6
List the points in a table.
Step 8
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 9