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Trigonometry Examples
Step 1
Combine and .
Step 2
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 tangent function, , for equal to to find where the vertical asymptote occurs for .
Step 3
Step 3.1
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 3.2
Divide each term in by and simplify.
Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
Step 3.2.2.1
Cancel the common factor of .
Step 3.2.2.1.1
Cancel the common factor.
Step 3.2.2.1.2
Divide by .
Step 3.2.3
Simplify the right side.
Step 3.2.3.1
Cancel the common factor of .
Step 3.2.3.1.1
Cancel the common factor.
Step 3.2.3.1.2
Divide by .
Step 4
Set the inside of the tangent function equal to .
Step 5
Step 5.1
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 5.2
Divide each term in by and simplify.
Step 5.2.1
Divide each term in by .
Step 5.2.2
Simplify the left side.
Step 5.2.2.1
Cancel the common factor of .
Step 5.2.2.1.1
Cancel the common factor.
Step 5.2.2.1.2
Divide by .
Step 5.2.3
Simplify the right side.
Step 5.2.3.1
Cancel the common factor of .
Step 5.2.3.1.1
Cancel the common factor.
Step 5.2.3.1.2
Rewrite the expression.
Step 6
The basic period for will occur at , where and are vertical asymptotes.
Step 7
Step 7.1
is approximately which is positive so remove the absolute value
Step 7.2
Multiply the numerator by the reciprocal of the denominator.
Step 7.3
Cancel the common factor of .
Step 7.3.1
Cancel the common factor.
Step 7.3.2
Rewrite the expression.
Step 8
The vertical asymptotes for occur at , , and every , where is an integer.
Step 9
Tangent only has vertical asymptotes.
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
Step 10