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Trigonometry Examples
Step 1
Step 1.1
For any , vertical asymptotes occur at , where is an integer. Use the basic period for , , to find the vertical asymptotes for . Set the inside of the cosecant function, , for equal to to find where the vertical asymptote occurs for .
Step 1.2
Solve for .
Step 1.2.1
Add to both sides of the equation.
Step 1.2.2
Multiply both sides of the equation by .
Step 1.2.3
Simplify both sides of the equation.
Step 1.2.3.1
Simplify the left side.
Step 1.2.3.1.1
Cancel the common factor of .
Step 1.2.3.1.1.1
Cancel the common factor.
Step 1.2.3.1.1.2
Rewrite the expression.
Step 1.2.3.2
Simplify the right side.
Step 1.2.3.2.1
Cancel the common factor of .
Step 1.2.3.2.1.1
Factor out of .
Step 1.2.3.2.1.2
Cancel the common factor.
Step 1.2.3.2.1.3
Rewrite the expression.
Step 1.3
Set the inside of the cosecant function equal to .
Step 1.4
Solve for .
Step 1.4.1
Move all terms not containing to the right side of the equation.
Step 1.4.1.1
Add to both sides of the equation.
Step 1.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.4.1.3
Combine and .
Step 1.4.1.4
Combine the numerators over the common denominator.
Step 1.4.1.5
Simplify the numerator.
Step 1.4.1.5.1
Multiply by .
Step 1.4.1.5.2
Add and .
Step 1.4.2
Multiply both sides of the equation by .
Step 1.4.3
Simplify both sides of the equation.
Step 1.4.3.1
Simplify the left side.
Step 1.4.3.1.1
Cancel the common factor of .
Step 1.4.3.1.1.1
Cancel the common factor.
Step 1.4.3.1.1.2
Rewrite the expression.
Step 1.4.3.2
Simplify the right side.
Step 1.4.3.2.1
Cancel the common factor of .
Step 1.4.3.2.1.1
Factor out of .
Step 1.4.3.2.1.2
Cancel the common factor.
Step 1.4.3.2.1.3
Rewrite the expression.
Step 1.5
The basic period for will occur at , where and are vertical asymptotes.
Step 1.6
Find the period to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.
Step 1.6.1
is approximately which is positive so remove the absolute value
Step 1.6.2
Multiply the numerator by the reciprocal of the denominator.
Step 1.6.3
Multiply by .
Step 1.7
The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.
Step 1.8
Cosecant only has vertical asymptotes.
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
Step 2
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Step 3
Since the graph of the function does not have a maximum or minimum value, there can be no value for the amplitude.
Amplitude: None
Step 4
Step 4.1
The period of the function can be calculated using .
Step 4.2
Replace with in the formula for period.
Step 4.3
is approximately which is positive so remove the absolute value
Step 4.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.5
Multiply by .
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
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Step 5.4
Cancel the common factor of .
Step 5.4.1
Factor out of .
Phase Shift:
Step 5.4.2
Cancel the common factor.
Phase Shift:
Step 5.4.3
Rewrite the expression.
Phase Shift:
Phase Shift:
Phase Shift:
Step 6
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: ( to the right)
Vertical Shift: None
Step 7
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Vertical Asymptotes: where is an integer
Amplitude: None
Period:
Phase Shift: ( to the right)
Vertical Shift: None
Step 8