4.1 Graphs of the Sine & Cosine Functions

A function f is periodic if f (x + h) = f (x) for every x in domain of f

Period of f = smallest positive number h

One cycle of graph is completed in each period

Ex 1) Verify that the graph represents a periodic function and identify its period.

Repeats in cycles periodic!Period = 4

Graph of y = sin x

0

6

4

3

2

3

22

• Period = 2π• For all x, sin (–x) = –sin x (odd function)• Symmetric wrt origin• Domain = R• Range = [–1, 1]• Zeros occur at multiples of π

01

22

.723

.8721

0

1

0

x y 1

–1

Graph of y = cos x

0

6

4

3

2

3

22

1

3.87

22

.721

20

1

0

1

x y 1

–1

• Period = 2π• For all x, cos (–x) = cos x (even function)• Symmetric wrt y-axis• Domain = R• Range = [–1, 1]• Zeros occur at odd multiples of

2

The sine and cosine functions are related to each other. They are called cofunctions.

Ex 1) Express each function in terms of its cofunction.

a)

sin cos and cos sin2 2

6 5sin cos cos cos

12 2 12 12 12 12

b) cos40 sin(90 40 ) sin50

We will now take a look at how we can transform the basic sine & cosine curves

Use Desmos app & the worksheet to help guide us.

Open Desmos. Choose , then Trigonometry , and then All the Trig Functions

Tap into box 7 and start deleting until all you are left with is box 2 sin (x)

We would like to adjust the window so that the x-axis is showing [–2π, 2π] and the y-axis is [–5, 5]Pinch & spread with 2 fingers to get the window just right

Now look at WS. A graph from [–2π, 2π] is pictured.We already know about parent graphs & transformations.Write down (and then share) what will happen to y = sin x if you graph #1 (and WHY).

Now enter #1: ½ sin (x) in box 3(to get ½, simply type 1 ÷ 2)

(to get sin, under tab and tab)

Was your guess right?!

Now, let’s repeat this process with #2 – 4.

Please don’t go to back side until we are all ready!!

On back of WS is y = cos x from [–2π, 2π].

We have just reminded ourselves & practiced graphing transformations using sine as the parent graph.

Let’s see how quickly (and accurately) you can graph the 4 transformations of y = cos x

On Your Mark….

All done! Quickly confirm with Desmos

We will do more involved transformations later in the chapter … today just the basics!

Get Set…. GO!!!

Homework

#401 Pg 187 #1, 2, 3, 7, 11, 14, 19–44 all omit 24, 25