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Lesson 1

1. Lesson 1

1.5. Explore

Mathematics 30-1 Module 5

Module 5: Trigonometry Applications and Identities

 

Explore

 

In Try This 1 you plotted the graph of a tangent function that should have looked like the first diagram that follows. The tangent function can be defined as the y-coordinate of the intersection of the terminal arm and a vertical line tangent to the right side of the unit circle as shown in the second and third diagrams.

 

This graph shows f at theta is equal to the tangent of theta where theta is greater than 110 degrees and less than 450 degrees.

 

These diagrams show the terminal arm intersecting a vertical tangent line. The distance from the x-axis to the intersection point is labelled tan theta.

 

When the terminal arm is in quadrant 1 or quadrant 4, the intersection of the terminal arm and the vertical tangent is used to find y.

These diagrams show the terminal arm intersecting a vertical tangent line. The distance from the x-axis to the intersection point is labelled tan theta.

 

When the terminal arm is in quadrant 2 or quadrant 3, an extension of the terminal arm is used to find the intersection with the vertical tangent.

 

In Try This 2 you will explore some characteristics of the tangent function.

 

Try This 2
  1. Using technology, graph the function y = tan x over the domain −4π< x < 4π.
  2. Sketch the graph of y = tan x. Draw in the vertical asymptotes.
  3.  
    1. Determine when tan x = 0.
    2. Write a general equation to represent all the real values of x that make tan x = 0.
  4. The graph of y = tan x is not defined for all x values and includes vertical asymptotes.
    1. At what x values do asymptotes occur?
    2. Write a general equation to represent all the asymptotes of y = tan x over the real numbers.
  5. Determine the period of y = tan x. How does this compare to the period of y = sin x or y = cos x?
  6. Determine the domain and range of y = tan x. How do these sets of values compare to the domain and range of y = sin x or y = cos x?
  7. Determine the amplitude of tan x. What problem do you notice when attempting to determine this amplitude?

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Your equation can be of the form x = __ + __ π, n ∈ I.
Your equation can be of the form x = ____ π, n ∈ I.