Period of Trigonometric Functions

In previous lessons we have learned how to translate any primary trigonometricfunction horizontally or vertically, and how to Stretch Vertically (change Amplitude).In this unit we are going to learn how to Stretch Horizontally. In most “Real World”applications that involve trigonometry, what we are doing is determining thetrigonometric function’s Period.

In order to master the techniques explained here it is very important that youundertake plenty of practice exercises so that they become second nature.

After reading this unit, you should be able to complete the following:

Determine the period for Sine, Cosine, and Tangent

Sketch the function on blank grid

Sketch the function on fixed grid

Recall:

The three basic trigonometric functions have periods as demonstrated below:

Sine function -> period is 2 radians or 360. Cosine function -> period is 2 radians or 360. Tangent function -> period is radians or 180.

The basic graphs of these 3 trigonometric functions are:

The length of one complete cycle of a trigonometric function is called the Period.Typically we use x=0 as the starting point for the graph. When the length of the periodis not the default, the functions will be written in the format similar to: sinf t kt ,

cosg x kx , or tanh k . Where the constant k aids us in determining the

period for the function.

The Period (wavelength) of sin(kt) and cos(kt)

If k>0, then the graph of sinf t kt or cosf t kt makes k complete cycles

between 0 and 2, and each functions has a period of:

Period=2k

Note: we are working in radians, else360

k

The Period of tan(kt)

If k>0, then the graph of tanf t kt makes k complete cycles between2 0

and2

, and each functions has a period of:

Period=k

Note: we are working in radians, else360

k

Example:

Determine the period of each of the functions:

a) sin 3y x

b) cos2t

f t

c) 3tan

2k

Solutions:

a) period =23

We have 3 complete cycles between 0 and 2

b) period =2 412

½ cycle between 0 and 2

c) period =2

3 32

Note: A calculator may not produce an accurate graph of trigonometric functionswith a large k value. For example the graph of sin 50f t t has 50

complete cycles between 0 and 2, but some calculators have problemsshowing this (try it).

Graphing The Sine and Cosine Trigonometric Functions by Hand

When we are asked to graph trigonometric functions by hand, two types of questionsare usually presented to you:

The first being that you have full control. A blank piece of graph paper is providedand you are to sketch your graph on the grid. You may be asked to one cycle or a setnumber of cycles, so use logic to guide you.

It is best to use 12 blank spaces or a multiple of 12 blank spaces for your cycle. Uselogic to determine which is best to provide the number of required cycles.

With 12 spaces (or multiple of 12) you can easily find the 4 intervals (or quarters) that

will correspond to the30, , , ,2

2 2

locations that would have a value of {-1, 0, 1}

values of either the sine or cosine function.

The second being that the grid with its domain is already provided for you. Here youhave to conform to its restrictions when graphing the function. There is a bit morework here, but a fun puzzle to solve.

Type 1: When you are given a blank grid.

These are the easiest to graph for you are given a blank grid and have full control, somake it easy for yourself.

If you are given no domain restrictions and you need to draw one complete cycle.

Step 1: Determine the PeriodStep 2: Let 12 spaces on the grid represent the period.Step 3: Determine the ¼ period, ½ period, ¾ period values for your function (these

will be 3 spaces apart on your grid).Step 4: Plot your sin or cos graph by using (0, 1, 0, -1, 0, 1, 0, -1) amplitude points

and follow the pattern.

If you need more than one cycle, see how many multiples of 12 you can use on yourgrid. Use logic to assist you, as you will need to break each cycle in to quarters, so youmay have to use 8 grid spaces, or 4 grid spaces per cycle.

Some questions may require you to plot both positive and neagative domain values. Ifthis is the case ensure you provide a symmetric axis location.

Example:

Graph sin 2f t t when given a blank grid

Solution:

Determine the period:22

Now horizontal grid markers will be at ¼ period, ½ period, ¾ period

1 1 3 3( 1) 2) 3)

4 4 2 2 4 4Q Q Q

Since this is a sine function. The f(t) values will (starting at t=0) follow the pattern {0,1, 0, -1, 0} at the quarter period t-values.

Plot the following 5 points: 30,0 , ,1 , ,0 , , 1 , ,0

4 2 4

Now connect the dots to complete the graph.

Example:

Graph cos10

y x

when given a blank grid.

Solution:

Determine the Period:2 102 20

10

Now horizontal grid markers will be at ¼ period, ½ period, ¾ period

1 1 31) 20 5 2) 20 10 3) 20 154 2 4

Q Q Q

Use these values to label your grid.

Since this is a cosine function. The y-values will (starting at x=0) follow the pattern{1, 0, -1, 0, 1} at the quarter period x-values. Plot the following 5 points:

0,1 , 5,0 , 10, 1 , 15,0 , 20,1

Type 2: When you are given a labelled grid.

These are harder to do; but by following the technique below you will find these areactually fun to plot. Remember the teacher or textbook will provide questions that fitnicely into the grid that they provide. This makes your job very easy.

If possible, try to provide 12 blank spaces for the cycle (12 is divisible by 2, 3, 4, 6and thus gives you more room to play). We will be using the 4 quarters of the cycle,thus if 12 spaces for each cycle does not fit into the grid, try 8, or even 4. Again havefun with the logic to make the graph fit into the set grid.

Step 1: Determine blank space width by domain divided by 12.

Step 2: Determine the period.

Step 3: If your teacher set up the question correctly, the period should be amultiple of blank grid squares. Count the numbers of grid squares for a

complete cycle. Useperiod

blank square width.

Step 4: Take Step 4’s value and divide by 4, this new result will be the numbers ofsquares each quarter of the function will take for the pattern (…, 0, 1, 0, -1, 0, 1, 0, …). Mark these quarter place markers on your grid starting at 0.

Step 5: Place the (…, 0, 1, 0, -1, 0, 1, 0, …) at each marked of spot on your grid,remember each quarter will be Step 4 number of spaces.

Step 6: You should label your grid at the ¼ period, ½ period, ¾ period values foryour function.

Example:

Graph sin 3y x when domain is 0 2x

Solution:

Our blank space width:2

12 12 6domain

The period of the graph is:23

The number of squares the period occupies is:

2period 2 63 4

1 black square width 36

The number of spaces per quarter is4 14 .

Therefore every 4 grid spaces the function will complete a full cycle, with 1 spaceproviding the quarter function point at the {… 0, 1, 0, -1, 0, …} pattern.

This gives points at: 2 1 2 2 2 3 2 40,0 , ,1 , ,0 , , 1 , ,0

3 4 3 4 3 4 3 4

Let’s graph it demonstrating the k=3 complete cycles.

Example

cos3xy

when domain is 2x and the grid is provided as below.

Solution:

Since we have 60 spaces on the horizontal axis, and we would prefer 12, we will use 5spaces per major tick.

Our blank space width :3

12 12 4domain

The period of the graph is:2 613

The number of squares a complete cycle occupies is:

period 6 46 24

space width4

Therefore every 24 grid spaces (with the grid only providing 12 spaces) the function

will complete a cycle with24

64 spaces providing the quarter function point interval

for the {… 0, 1, 0, -1, 0, …} pattern.

In this question we will only be able to graph 1 point (the ¼ period value).

Since this is a cosine function, start at x=0 with (0,1) then its next point will be 6 grid

space from 0 at the location3

6 ,0 ,04 2

.

Let’s graph it.

Example

Graph 15sin6

f for

3 32 2 on the following grid.

Solution:

The grid spacing does not look nice. We have 18 major horizontal tick marks and5 space-per-tick mark (giving a total of 90 horizontal spaces). Let’s first find theperiod of our graph.

Period =2 6 12 4

215 15 15 56

Now since we have 18 horizontal ticks over a domain of 3, the width of each

horizontal tick is318 6 .

Oh no, it looks like we have a problem. The period of45

does not work well with

the horizontal tick spacing of6

. We could use our calculator to approximate the

locations, or remember that the question is set up to work on the given grid(most times). What about those 5 space-per-tick mark for a total of 90 spaces,can we use them?

The width of each space is390 30

Now how many space does it take for the period of45

?

Number of spaces per cycle =

4period 4 305 24

space width 530

Hey, this is a nice number to work with.

Each quarter cycle has a length of24 64 spaces.

Since this is a Sine Function, it starts at (0,0) and each 6 spaces will follow the

usual Sine pattern. We can use the horizontal tick width of6

to place labels on

the graph for reference to improve readability.