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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 secant function, , for equal to to find where the vertical asymptote occurs for .
Step 1.2
Solve for .
Step 1.2.1
Move all terms not containing to the right side of the equation.
Step 1.2.1.1
Subtract from both sides of the equation.
Step 1.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.2.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.2.1.3.1
Multiply by .
Step 1.2.1.3.2
Multiply by .
Step 1.2.1.4
Combine the numerators over the common denominator.
Step 1.2.1.5
Simplify the numerator.
Step 1.2.1.5.1
Multiply by .
Step 1.2.1.5.2
Subtract from .
Step 1.2.1.6
Move the negative in front of the fraction.
Step 1.2.2
Divide each term in by and simplify.
Step 1.2.2.1
Divide each term in by .
Step 1.2.2.2
Simplify the left side.
Step 1.2.2.2.1
Cancel the common factor of .
Step 1.2.2.2.1.1
Cancel the common factor.
Step 1.2.2.2.1.2
Divide by .
Step 1.2.2.3
Simplify the right side.
Step 1.2.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.2.3.2
Multiply .
Step 1.2.2.3.2.1
Multiply by .
Step 1.2.2.3.2.2
Multiply by .
Step 1.3
Set the inside of the secant 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
Subtract from 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
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.4.1.3.1
Multiply by .
Step 1.4.1.3.2
Multiply by .
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
Subtract from .
Step 1.4.2
Divide each term in by and simplify.
Step 1.4.2.1
Divide each term in by .
Step 1.4.2.2
Simplify the left side.
Step 1.4.2.2.1
Cancel the common factor of .
Step 1.4.2.2.1.1
Cancel the common factor.
Step 1.4.2.2.1.2
Divide by .
Step 1.4.2.3
Simplify the right side.
Step 1.4.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.4.2.3.2
Multiply .
Step 1.4.2.3.2.1
Multiply by .
Step 1.4.2.3.2.2
Multiply by .
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.6.2
Cancel the common factor of .
Step 1.6.2.1
Cancel the common factor.
Step 1.6.2.2
Divide 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
Secant 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
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.4
Cancel the common factor of .
Step 4.4.1
Cancel the common factor.
Step 4.4.2
Divide 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
Multiply .
Step 5.4.1
Multiply by .
Phase Shift:
Step 5.4.2
Multiply by .
Phase Shift:
Phase Shift:
Phase Shift:
Step 6
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: ( to the left)
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 left)
Vertical Shift: None
Step 8