4 6 graphs of other trigonometric functions
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4.6 Graphs of Other Trigonometric Functions. Objectives Understand the graph of y = tan x Graph variations of y = tan x Understand the graph of y = cot x Graph variations of y = cot x Understand the graphs of y = csc x and y = sec x Pg. 531 #2-46 (every other even). y = tan x.

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4.6 Graphs of Other Trigonometric Functions

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4 6 graphs of other trigonometric functions

4.6 Graphs of Other Trigonometric Functions

  • Objectives

    • Understand the graph of y = tan x

    • Graph variations of y = tan x

    • Understand the graph of y = cot x

    • Graph variations of y = cot x

    • Understand the graphs of y = cscx and y = sec x

      Pg. 531 #2-46 (every other even)


Y tan x

y = tan x

  • Going around the unit circle, where the y value is 0, (sin x = 0), the tangent is undefined.

  • At x = the graph of y = tan x has vertical asymptotes

  • x-intercepts where cos x = 0, x =


Characteristics of y tan x

Characteristics of y = tan x

  • Period =

  • Domain: (all reals except odd multiples of

  • Range: (all reals)

  • Vertical asymptotes: odd multiples of

  • x – intercepts: all multiples of

  • Odd function (symmetric through the origin, quad I mirrors to quad III)


Graphing y a tan bx c

Graphing y = A tan (Bx – C)

  • Find two consecutive asymptotes by finding an interval containing one period. A pair of consecutive asymptotes occur at

    • and

  • Identify an x-intercept midway between the consecutive asymptotes.

  • Find the points on the graph at and of the way between the consecutive asymptotes. These points will have y-coordinates of –A and A.

  • Use steps 1-3 to graph one full period of the function. Add additional cycles to the left and right as needed.


  • 4 6 graphs of other trigonometric functions

    1. Graph y = 3 tan 2x for –π ∕4 <x< 3π∕4


    4 6 graphs of other trigonometric functions

    2. Graph two full periods of tan(x - π∕2)


    Graphing y cot x

    Graphing y = cot x

    • Vertical asymptotes are where sin x = 0, (multiples of pi)

    • x-intercepts are where cos x = 0 (odd multiples of pi/2)


    4 6 graphs of other trigonometric functions

    • Graphing y = A cot (Bx-C)

    • Find two consecutive asymptotes by finding a pair.

    • One pair occurs at: Bx-C = 0 and Bx-C = π

    • Identify an x-intercept, midway between the consecutive asymptotes.

    • Find the points and of the way between the consecutive asymptotes. These points have y-coordinates of A and –A.

    • Use steps 1-3 to graph one full period of the function. Add additional cycles to the left and right as needed.


    4 6 graphs of other trigonometric functions

    3. Graph y = (1 ∕ 2) cot (π∕2) x


    Y csc x

    y = csc x

    • Reciprocal of y = sin x

    • Vertical tangents where sin x = 0 (x = integer multiples of pi)

    • Range:

    • Domain: all reals except integer multiples of pi

    • Graph on next slide

    Take notice of the blue boxes on page 527. The graphs demonstrate the close relationships between sine and cosecant graphs.


    Graph of y csc x

    Graph of y = csc x


    4 6 graphs of other trigonometric functions

    4. Use the graph of y = sin (x + π∕4) to obtain the graph of y = csc (x + π∕4)


    Y sec x

    y = sec x

    • Reciprocal of y = cos x

    • Vertical tangents where cos x = 0 (odd multiples of pi/2)

    • Range:

    • Domain: all reals except odd multiples of pi/2

    • Graph next page

    Again, take notice of the blue boxes on page 527. The graphs demonstrate the close relationships between cosine and secant graphs.


    Graph of y sec x

    Graph of y = sec x


    4 6 graphs of other trigonometric functions

    5. Graph y = 2 sec 2x for -3π∕4 <x< 3π∕4


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