Graphs of Tangent & Cotangent

Today we will graph tangent and cotangent curves using our

knowledge of sine and cosine curves and also rational functions.

The graph of the tangent function is shown below.

As with the sine and cosine the graph tells us quite a bit about the function’s properties.

Where do the asymptotes occur for tangent?

What is the period of tangent?

What is the domain of tan x?

What is the range of tan x?

We can analyze why the graph of tangent behaves the way it does:

It follows from the definitions of the trigonometric functions that

Unlike the sine and cosine, the tangent function has a denominator that might be zero, which makes the function undefined.

sintan

cos

y ttx t

Not only does this actually happen, it happens an infinite number of times.

An asymptote is an indication of a place where the function is undefined. Where do the asymptotes occur for the tangent? Why do you think they occur where they do?

The tangent function has asymptotes where the cosine is zero.

The tangent function has points of inflection (crossings) where the sine function is zero.

What about the period of y = tan x?

Let’s go back to the unit circle and look at the how the tangent values wrap around the circle:

( , tan )

3( , )6 3

( ,1)4

( , 3)3

( , )22( , 3)33( , 1)4

5 3( , )6 3( ,0)

x x

undef

7 3( , )6 35( ,1)44( , 3)33( , )25( , 3)37( , 1)4

11 3( , )6 3

(2 ,0)

undef

Think about the fact that tangent is sin x/ cos x and the signs of the quadrants as we go around the unit circle – the tangent repeats itself after just ½ way around the circle. Therefore the period for tangent is π.

Important Characteristics of the Tangent

Domain: all real numbers except odd multiples of These are asymptotes.

*This is where the cosine is zero.

Range:

Period: (new period: )

Inflection points: Halfway between the vertical asymptotes will be a crossing.Remember this is where the sine is zero.

The graph of tangent increases from left to right

2

( , )

B

“Flipping” (Reflecting) and Amplitude

y=tan x has the same flipping characteristics as y = sin x. Do NOT flip until the very end. If this graph is flipped it decreases from left to right.

Tangent has no defined amplitude, since the graph increases (or decreases) without bound.

Tangent Example

Determine new x-values(use B & C) Shift up or down (use D) Graph denominator function Draw in vertical asymptotes Does it flip? Graph one period

tan[ 2( )] 14

y x

sin(tan )

cos

xx

x

Answer to Tangent Example

What about Cotangent?

As with the sine and cosine the graph tells us quite a bit about the function’s properties.

Where do the asymptotes occur for cotangent?

What is the period of cotangent?

What is the domain of cot x?

What is the range of cot x?

The cotangent has asymptotes where the sine function is zero.

The cotangent has points of inflection (crossings) where the cosine function is zero.

Important Characteristics of the Cotangent

Domain: all real numbers except multiples of These are asymptotes. *This is where the sine is zero.

Range:

Period: (new period: )

Inflection points: Halfway between the vertical asymptotes will be a crossing.Remember this is where the cosine is zero.

The graph of cotangent decreases from left to right

( , )

B

“Flipping” (Reflecting) and Amplitude

y=cot x also has the same flipping characteristics as y = sin x. Do NOT flip until the very end. If this graph is flipped it increases from left to right.

Cotangent has no defined amplitude, since the graph increases (or decreases) without bound.

Cotangent Example

Determine new x-values(use B & C) Shift up or down (use D) Graph denominator function Draw in vertical asymptotes Does it flip? Graph one period

cos(cot )

sin

xx

x

4cot( ) 12

y x

Answer to Cotangent Example

Try These:

tan( )4

tan(2 )

4cot( ) 12

tan[2( )] 14

cot[3( )] 112

y x

y x

y x

y x

y x

Assignment

A 1.4 Sect II(Pg. 401 #13 – 16)

A 1.4 Sect III

See you tomrrw!