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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
Multiply the numerator by the reciprocal of the denominator.
Step 3.5
Multiply 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:
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
Divide by .
Step 6.1.2.2
Add and .
Step 6.1.2.3
The exact value of is .
Step 6.1.2.3.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 6.1.2.3.2
Apply the sine half-angle identity.
Step 6.1.2.3.3
Change the to because sine is positive in the first quadrant.
Step 6.1.2.3.4
Simplify .
Step 6.1.2.3.4.1
The exact value of is .
Step 6.1.2.3.4.2
Multiply by .
Step 6.1.2.3.4.3
Subtract from .
Step 6.1.2.3.4.4
Divide by .
Step 6.1.2.3.4.5
Rewrite as .
Step 6.1.2.3.4.6
Pull terms out from under the radical, assuming positive real numbers.
Step 6.1.2.4
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
Divide by .
Step 6.2.2.2
Add and .
Step 6.2.2.3
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
Divide by .
Step 6.3.2.2
Add and .
Step 6.3.2.3
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
Divide by .
Step 6.4.2.2
Add and .
Step 6.4.2.3
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
Divide by .
Step 6.5.2.2
Add and .
Step 6.5.2.3
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 left)
Vertical Shift: None
Step 8