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 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) 1.Find two consecutive asymptotes by finding an interval containing one period. A pair of consecutive asymptotes occur at and 2.Identify an x-intercept midway between the consecutive asymptotes. 3.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. 4.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

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

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) 1.Find two consecutive asymptotes by finding a pair. One pair occurs at: Bx-C = 0 and Bx-C = π 2.Identify an x-intercept, midway between the consecutive asymptotes. 3.Find the points and of the way between the consecutive asymptotes. These points have y-coordinates of A and –A. 4.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.

Graph of y = csc x

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.

Graph of y = sec x

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