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# 4.6 Graphs of Other Trigonometric Functions - PowerPoint PPT Presentation

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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• 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)

• 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 =

• 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)

• 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.

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

• 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

• 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)

• 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