Trigonometry Examples

Find the asymptotes.
Tap for more steps...
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 cotangent function, bx+c, for equal to to find where the vertical asymptote occurs for .
Divide each term by and simplify.
Tap for more steps...
Divide each term in by .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Cancel the common factor.
Divide by to get .
Divide by to get .
Set the inside of the cotangent function equal to .
Divide each term by and simplify.
Tap for more steps...
Divide each term in by .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Cancel the common factor.
Divide by to get .
The basic period for will occur at , where and are vertical asymptotes.
The absolute value is the distance between a number and zero. The distance between and is .
The vertical asymptotes for occur at , , and every , where is an integer.
Cotangent 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 n is an integer
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Since the graph of the function does not have a maximum or minimum value, there can be no value for the amplitude.
Amplitude: None
Find the period using the formula .
Tap for more steps...
The period of the function can be calculated using .
Period:
Replace with in the formula for period.
Period:
The absolute value is the distance between a number and zero. The distance between and is .
Period:
Period:
Find the phase shift using the formula .
Tap for more steps...
The phase shift of the function can be calculated from .
Phase Shift:
Replace the values of and in the equation for phase shift.
Phase Shift:
Divide by to get .
Phase Shift:
Phase Shift:
Find the vertical shift .
Vertical Shift:
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: ( to the right)
Vertical Shift:
Select a few points to graph.
Tap for more steps...
Find the point at .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and to get .
The exact value of is .
Multiply by to get .
The final answer is .
Find the point at .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and to get .
The exact value of is .
Multiply by to get .
The final answer is .
Find the point at .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and to get .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the second quadrant.
The exact value of is .
Multiply by to get .
Multiply by to get .
The final answer is .
Find the point at .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and to get .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
The exact value of is .
Multiply by to get .
The final answer is .
Find the point at .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and to get .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.
The exact value of is .
Multiply by to get .
Multiply by to get .
The final answer is .
List the points in a table.
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Vertical Asymptotes: where n is an integer
Amplitude: None
Period:
Phase Shift: ( to the right)
Vertical Shift:
Enter YOUR Problem

Enter the email address associated with your Mathway account below and we'll send you a link to reset your password.

Please enter an email address
Please enter a valid email address
The email address you entered was not found in our system
The email address you entered is associated with a Facebook user
We're sorry, we were unable to process your request at this time

Mathway Premium

Step-by-step work + explanations
  •    Step-by-step work
  •    Detailed explanations
  •    No advertisements
  •    Access anywhere
Access the steps on both the Mathway website and mobile apps
$--.--/month
$--.--/year (--%)

Mathway Premium

Visa and MasterCard security codes are located on the back of card and are typically a separate group of 3 digits to the right of the signature strip.

American Express security codes are 4 digits located on the front of the card and usually towards the right.
This option is required to subscribe.
Go Back

Step-by-step upgrade complete!

Mathway requires javascript and a modern browser.
  [ x 2     1 2     π     x d x   ]