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.
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
 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
 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
 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) = 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
 Click on the radio button of cot(x). The red broken lines are the
vertical asymptotes for the graph of cot(x). Use the fact that
cot(x) = 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
 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) = 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
 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) = 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
 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.
 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)?
 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)?
 Use the results of part 1 above to express 5 of the trigonometric functions in terms of the cosine function only?
 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.
