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4.6 Graphs of Other Trigonometric FunctionsPowerPoint Presentation

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 = cscx and y = sec x
Pg. 531 #2-46 (every other even)

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

- 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) 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.

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

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

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)

- 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.

3. Graph y = (1 ∕ 2) cot (π∕2) 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.

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

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.

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

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