Trigonometry Examples

Find the Asymptotes y=2csc(4x)
Step 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 2
Divide each term in by and simplify.
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Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Divide by .
Step 2.3
Simplify the right side.
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Step 2.3.1
Divide by .
Step 3
Set the inside of the cosecant function equal to .
Step 4
Divide each term in by and simplify.
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Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
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Step 4.2.1
Cancel the common factor of .
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Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
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Step 4.3.1
Cancel the common factor of and .
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Step 4.3.1.1
Factor out of .
Step 4.3.1.2
Cancel the common factors.
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Step 4.3.1.2.1
Factor out of .
Step 4.3.1.2.2
Cancel the common factor.
Step 4.3.1.2.3
Rewrite the expression.
Step 5
The basic period for will occur at , where and are vertical asymptotes.
Step 6
Find the period to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.
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Step 6.1
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2
Cancel the common factor of and .
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Step 6.2.1
Factor out of .
Step 6.2.2
Cancel the common factors.
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Step 6.2.2.1
Factor out of .
Step 6.2.2.2
Cancel the common factor.
Step 6.2.2.3
Rewrite the expression.
Step 7
The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.
Step 8
Cosecant only has vertical asymptotes.
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
Step 9