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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 tangent function, , for equal to to find where the vertical asymptote occurs for .
Step 1.2
Divide each term in by and simplify.
Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
Step 1.2.2.1
Cancel the common factor of .
Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
Step 1.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.3.2
Cancel the common factor of .
Step 1.2.3.2.1
Move the leading negative in into the numerator.
Step 1.2.3.2.2
Factor out of .
Step 1.2.3.2.3
Cancel the common factor.
Step 1.2.3.2.4
Rewrite the expression.
Step 1.2.3.3
Move the negative in front of the fraction.
Step 1.3
Set the inside of the tangent function equal to .
Step 1.4
Divide each term in by and simplify.
Step 1.4.1
Divide each term in by .
Step 1.4.2
Simplify the left side.
Step 1.4.2.1
Cancel the common factor of .
Step 1.4.2.1.1
Cancel the common factor.
Step 1.4.2.1.2
Divide by .
Step 1.4.3
Simplify the right side.
Step 1.4.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.4.3.2
Cancel the common factor of .
Step 1.4.3.2.1
Cancel the common factor.
Step 1.4.3.2.2
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.
Step 1.6.1
is approximately which is positive so remove the absolute value
Step 1.6.2
Cancel the common factor of .
Step 1.6.2.1
Cancel the common factor.
Step 1.6.2.2
Rewrite the expression.
Step 1.7
The vertical asymptotes for occur at , , and every , where is an integer.
Step 1.8
Tangent 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
Cancel the common factor of .
Step 4.4.1
Cancel the common factor.
Step 4.4.2
Rewrite the expression.
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: None
Period:
Phase Shift: None
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: None
Vertical Shift: None
Step 8