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 =

,...)2

3,2,

2,

2

3(...

,...)2,,0,,2(...

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)

2

Graphing y = A tan (Bx – C)

2Bx 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.

2Bx C

1

4

3

4

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.

1

4

3

4

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

),1[]1,(

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

),1[]1,(

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