Cosecant Function

A tutorial on exploring the cosecant function given by

f(x) = a * csc( b x + c ) + d


and its period, phase shift, asymptotes domain and range is presented. An interactive applet is used where parameters a, b, c and d are changed and their effects explored.

Interactive Tutorial

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1 - Click on the button above "click here to start" and maximize the window obtained.
  1. 2 - Use the scroll bar to set a = 1, b = 1, c = 0 and d = 0. Now change a , how does it affect the graph? Does it affect the range? If yes how?

  2. 3 - Set a = 1,c = 0,d = 0 and change b. For each value of b find the period from the graph and compare it to 2pi/|b|. How does b affect the graph of f(x)? How does it affect the asymptotes?

  3. 4 - Set a = 1,b = 1,d = 0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to - c / b.

  4. 5 - Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to - c / b.

  5. 6 - Repeat 4 and 5 above for b = 2, 3 and 4.

  6. 7 - Set a,b and c to non zero values and change d. What is the direction of the shift of the graph?

  7. 8 - Which of the parameters affect the positions of the asymptotes (in red)? Explain analytically.

  8. 8 - Which of the parameters affect the domain of the cosecant function? Explain analytically.

  9. 9 - Which of the parameters affect the range of the cosecant function? Explain analytically.


More references on the trigonometric functions.
Trigonometric Functions.

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