Warm-up: Complete the table from = /2 to = /2 (one one interval), with the given 5 input angle values. x sinx cosx /2 1 0 und 2 2 2 2 1 0 0 1 0 /4 2 2

2 2 1 0 und /4 /2 1 y = tanx HW: Graph Tangent and Cotangent Functions Objective: Graph tangent and cotangent functions. f x tan x f x cot x Tangent Function sin Recall that tan cos undefined. when cos = 0, tan is This occurs at intervals, offset by /2: { /2, /2, 3/2, 5/2, } From the warm-up we can create a t-table for y = tanx

x sinx cosx y = tanx x y = tanx /2 1 0 und /2 und 2 2 2 2 1 /4 1 0 0

1 0 0 0 /4 2 2 2 2 1 /4 1 /2 1 0 und /2 und /4 We are interested in the graph of y = tan x y

x y = tan x vertical asymptote undefined 2 1 4 0 0 4 1 2 x undefined 2 4

4 2 Graph of the Tangent Function y Properties of y = tan x 1. Domain : all real x x k , k Z 2 2. Range: (, +) 3. Period: / 4. Vertical asymptotes: x k , k Z 2 2 3 2 2 period: 3 2 x Transformations apply as usual. Lets try one. up 2

y 2 tan x 4 reflect about x-axis y tan x right /4 y 2 tan x 4 y tan x y tan x 4 Finding Domain and Range: y 2 tan x 4 1. Domain for y = tanx: all real x x k , k Z

2 For y 2 tan x 4 x k , k Z 4 2 3 Domain : x k ,k Z 4 2. Range: (, +) Finding the Period: T Since the period of tangent is , the period of tan x is: The period would be /2 y tan 2 x y = tan x y y = tan 2x y 2 3

2 3 2 x 2 2 period: 4 4 2 period: / 2 x Graphing Variations of y = tan x and distance from asymptote to asymptote Properties of y = tan x 1. Domain : all real x x k , k Z 2 2. Range: (, +)

3. Period: / 4. Vertical asymptotes: x k , k Z 2 Example: Graphing a Tangent Function Graph y = 3 tan 2x and distance from asymptote to asymptote Step 1: Find two asymptotes. Bx C 2 2 2x 2 2 x Bx C 2 x 4 x 4

An interval containing one period is 4 4 , . 4 4 or / = /2 Graph y = 3 tan 2x and distance from asymptote to asymptote Step 2: Identify an x-intercept, midway between the consecutive asymptotes. x = 0 is midway between and The graph passes through (0, 0).4 . 4

4 4 Graph y = 3 tan 2x Step 3: Find points on the graph 1/4 and 3/4 of the way between the consecutive asymptotes. These points have y-coordinates of A and A. and distance from asymptote to asymptote The graph passes through , 3 and ,3 . 8 8 4 8 8 4

Finding Domain and Range: y = 3 tan 2x Properties of y = tan x 1. Domain : all real x x k , k Z 2 2. Range: (, +) 3. Period: / 4. Vertical asymptotes: x k 4 8 8 4 , kZ 2 Properties for y = 3tan2x ,k Z 2 k

x ,k Z 2 4 Domain: 2 x k Range: (, +) Period : 2 Vertical asymptotes: x k , kZ 2 4 Cotangent Function Recall that cot cos . sin when sin = 0, cot is undefined. This occurs at intervals, starting at 0: { , 0, , 2, } Lets create a t-table from = 0 to = (one one interval), with 5 input angle values. 0 sin cos

cot cot 0 Und 0 1 Und /4 2 2 2 2 1 /4 1 /2 1 0 0 /2

0 3/4 2 2 1 3/4 1 Und 0 2 2 1 Und Graph of Cotangent Function: Periodic Vertical asymptotes where sin = 0 cos cot sin cot

cot 0 und /4 1 /2 0 3/4 1 und 3/2 - /2 /2 3/2 Graph of the Cotangent Function cos x To graph y = cot x, use the identity cot x . sin x

At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y Properties of y = cot x y cot x 1. Domain : all real x x k , k Z 2. Range: (, +) 3. Period: / 4. Vertical asymptotes: x k , k Z vertical asymptotes x 3 2 2 x 2 x 0 3 2 2 x

x 2 Graphing Variations of y = cot x Properties of y = cot x 1. Domain : all real x x k , k Z 2. Range: (, +) 3. Period: / 4. Vertical asymptotes: x k , k Z and distance from asymptote to asymptote Example: Graphing a Cotangent Function 1 1 y cot x 2 2 Bx C 0 Bx C 1 x 0 2 1 x 2 x 0 y x 2 x 3

2 2 2 3 2 2 1 1 Graph y cot x 2 2 Step 2 Identify an x-intercept midway between the consecutive asymptotes. x = is midway between x = 0 and x = 2. The graph passes through (, 0). y x 3 2 2 2 3 2

2 1 1 Graph y 2 cot 2 x Step 3 Find points on the graph 1/4 and 3/4 of the way between consecutive asymptotes. These points have y-coordinates of A and A. The graph passes through 3 , 1 and 1 , . 2 2 2 2 y x 3 2 2 2 2 3 2 Finding Domain and Range:

1 1 1 1 y cot x Properties for y cot x 2 2 1 2 2 x k , k Z Domain: Properties of y = cot x 1. Domain : all real x x k , k Z 2. Range: (, +) 3. Period: / 4. Vertical asymptotes: x k , k Z 2 x 2k , k Z Range: (, +) y Period : 2 Vertical asymptotes: x 2k , k Z x 3

2 2 2 2 3 2 and distance from asymptote to asymptote Sneedlegrit: HW: Graph Tangent and Cotangent Functions