Graphs of Basic Trigonometric Functions

The graphs and properties such as domain, range, vertical asymptotes and zeros of the 6 basic trigonometric functions: \( \sin (x) \) , \( \cos (x) \) , \( \tan(x) \), \( \cot (x) \) , \( sec (x) \) and \( \csc (x) \) are explored using an html 5 applet.

Once you finish the present tutorial, you may want to go through a self test on trigonometric graphs. If needed, Free graph paper is available.

\( f(x) = \sin(x) \)    \( g(x)= \csc(x) \)    \( h(x) = \cos(x) \)    \( i(x) = \sec(x) \)    \( j(x) = \tan(x) \)    \( k(x) = \cot(x) \)

Note: Vertical asymtotes, if any, are shown as red dashed lines.

click on the button above "plot" to start investigating the graphs of the 6 basic trigonometry functions and their properties. Select a function and plot. Vertical asymptotes are shown as red dashed lines.

TUTORIAL (1) - Domain, Range, Zeros and Vertical Asymptotes of the 6 Basic Trigonometric Functions

  1. Click on the radio button of \( \sin (x) \) and use the graph to determine the range of \( \sin (x) \). What is the domain of \( \sin (x) \)? What are the zeros of \( \cos(x) \)?

    Answer

  2. Click on the radio button of \( \cos (x) \) and use the graph to determine the range of \( \cos (x) \). What is the domain of cos (x)? What are the zeros of \( \cos(x) \)?

    Answer

  3. Click on the radio button of \( \tan (x) \). The red broken lines are the vertical asymptotes for the graph of \( \tan (x) \). Use the identity \[ tan(x) = \dfrac{\sin(x)}{\cos(x)} \] to find the domain of tan(x). (Hint: find the zeros of the denominator and exclude them from the set of real numbers). The same zeros of the denominator gives you equations of the vertical asymptotes. Find the vertical asymptotes (red dashed lines) and the zeros of \( \tan(x) \). Now use the graph of tan (x) to check your answers, domain and vertical asymptotes, found analytically. Use the fact that vertical asymptotes means an increases or decreases without bound to find the range of tan(x).

    Answer

  4. Click on the radio button of \( \cos (x) \). The red broken lines are the vertical asymptotes for the graph of \( \cot (x) \). Use the fact that \[ cot(x) = \dfrac{\cos(x) }{\sin(x) } \] to find the domain of cot(x). (Hint: find the zeros of the denominator and exclude them from the set of real numbers). The same zeros of the denominator gives you equations of the vertical asymptotes. Find the vertical asymptotes and the zeros of cot(x). Now use the graph of cot(x) to check your answers, domain and vertical asymptotes, found analytically. Use the concepts of vertical asymptotes to determine the range of cot(x).

    Answer

  5. Click on the radio button of \( \sec(x) \). The red broken lines are the vertical asymptotes for the graph of sec(x).  Use the fact that \[ \sec(x) = \dfrac{1}{\cos(x)} \] to find the domain of sec(x) and vertical asymptotes for the graph of \( \sec(x) \). Now use the graph of \( \sec(x) \) to check your answers, domain and vertical asymptotes, found analytically. Use the concepts of vertical asymptotes to determine the range of sec(x).

    Answer

  6. Click on the radio button of csc(x). The red broken lines are the vertical asymptotes for the graph of csc(x).  Use the fact that \[ \csc(x) = \dfrac{1}{\sin(x)} \] to find the domain of \( \csc(x) \). (Hint: find the zeros of the denominator and exclude them). The same zeros of the denominator gives you equations of the vertical asymptotes. Find the vertical asymptotes. Now use the graph of csc(x) to check your answers, domain and vertical asymptotes, found analytically. Use the concepts of vertical asymptotes to determine the range of \( \csc(x) \).

TUTORIAL (2) - Relationship Between Basic Trigonometric Functions

  1. Compare the graphs of \( \sin(x) \) and \( \cos(x) \) and express \( \sin(x) \) in term of a cosine function and \( \cos(x) \) in term of a sine function using properties of shifting.
  2. Compare the graphs of \( \sin(x) \) and \( \csc(x) \)? Why are the zeros of sin(x) at the same location as the vertical asymptotes of \( \csc(x) \)?
  3. Compare the graphs of \( \cos(x) \) and \( \sec(x) \) ? Why are the zeros of \( \cos(x) \) at the same location as the vertical asymptotes of \( \sec(x) \) ?
  4. Use the results of part 1 above to express 5 of the trigonometric functions in terms of the cosine function only?
  5. Use the results of part 1 above to express 5 of the trigonometric functions in terms of sine function only?

Answer

More references and links related to trigonometric functions and their properties.