Starting off the day with a problem…The Bolivar Lighthouse is located on a small island 350ft fromthe shore of the mainland as shown in the figure.

x d350

ft

(a) Express the distance d as a functionof the angle x.

350cos xd

350secd x

(b) If x is 1.55 rad, what is d ?

350sec 1.55d 16831.108 ft

And another one just for good measureA hot-air balloon is being blown due east from point P andtraveling at a constant height of 800ft. The angle y is formed bythe ground and the line of vision P to the balloon. This anglechanges as the balloon travels.

xy

800

ft800tan yx

P

(a) Express the horizontal distance xas a function of the angle y.

800cotx y (b) When the angle is rad, what is its horizontal distancefrom P ?

20 800cot 20x 5051.001 ft

(c) An angle of rad is equivalent to how many degrees?2020 9

GRAPHS OF COMPOSITE

TRIGONOMETRIC FUNCTIONSThe beginning of

Section 4.6a

Example 1: Combining the sine function with2x

Graph each of the following functions for ,adjusting the vertical window as needed. Which of the functionsappear to be periodic?

2 2x

21 siny x x

22 siny x x

23 siny x

24 siny xFirst, put all four functions into your calculator, then you can turntheir graphs on and off as needed…

Vertical window

[–10, 20]

[–25, 25]

[–1.5, 1.5]

[–1.5, 1.5]

Only the graph of thisthird function exhibitsperiodic behavior overthe given interval!!!

Example 2: Verifying periodicity algebraically

Verify algebraically that is periodic anddetermine its period graphically.

2sinf x x

2

22 sin 2f x x It follows that

First, recall the period of the basic sine function

sin 2 sinx x Next, recall the fact that for all x.(Can you explain why this is true???)

2sin x f xBy the periodicity of sine

Check the graph What does the period appear to be?

Period =

More Guided PracticeGraph the given functions for , adjusting thevertical window as needed. State whether or not the functionsappears to be periodic.

2 ,2 by 5,20 2 2cosf x x x 1.Graph window:

Not Periodic

2 2x

2 ,2 by 12,12 2 cosf x x x2.Graph window:

Not Periodic

2 ,2 by 40,40 22cos 4f x x 3.Graph window:

Periodic

More Guided PracticeVerify algebraically that the given function is periodic anddetermine its period graphically. Sketch a graph showing twoperiods.

2 3cosf x x1.

Since the period of cos(x) is , we have

32 cos 2f x x 3cos x f x

2 ,2 by 1.5,1.5 Graph window:

More Guided PracticeVerify algebraically that the given function is periodic anddetermine its period graphically. Sketch a graph showing twoperiods.

2 3cosf x x2.

Since the period of cos(x) is , we have

32 cos 2f x x 3cos x f x

, by 1,2 Graph window:

Practice ProblemsProve algebraically that is periodic and findthe period graphically. State the domain and range and sketcha graph showing two periods.

3sinf x x

First, a reminder note regarding notation:

3sinf x xA function like is more frequently written as

3sinf x x(but this shorthand notation will not be recognized by a calculator)

Practice ProblemsProve algebraically that is periodic and findthe period graphically. State the domain and range and sketcha graph showing two periods.

3sinf x x

2f x f x To prove that the function is periodic, we need to show that

32 sin 2f x x for all x.

3sin 2x Changing notation

3sin x By periodicity of sine3sin x Changing notation f x

Practice ProblemsProve algebraically that is periodic and findthe period graphically. State the domain and range and sketcha graph showing two periods.

3sinf x x

2 ,2 by 1.5,1.5 Graph the function in the window:

2What does the period appear to be? Period =

, Domain:

1,1Range:

How does the graph of this functioncompare to that of the basic sinefunction??? (let’s graph both in thesame window)

Practice ProblemsFind the domain, range, and period of each of the followingfunctions. Sketch a graph showing four periods.

2 ,2 by 1.5,5 Graph window:

Wherever the basic tangent function is defined, f is also defined.

Domain: All reals except odd multiples of

tanf x x

2The range of the basic tangent function is all reals, but f is alwaysgreater than or equal to zero. (why???) Range: 0,

The period of f is the same as the basic tangent function:

Practice ProblemsFind the domain, range, and period of each of the followingfunctions. Sketch a graph showing four periods.

2 ,2 by 1,3 Graph window:

Wherever the basic sine function is defined, g is also defined.

sing x x

Range: 0,1The period of g is half that of the basic sine function:

, Domain:

The range of the basic sine function is –1 to 1 (inclusive), but g isalways greater than or equal to zero.

Practice ProblemsThe graph of oscillates between twoparallel lines. What are the equations of the two lines?

The basic sine function oscillates between –1 and 1, so f(x)must oscillate between 0.5x – 1 and 0.5x + 1…

0.5 sinf x x x

0.5 1y x The lines: 0.5 1y x

2 ,2 by 4,4 To verify our answer visually, graph all three functions in

Bonus question: Is f(x) periodic? Why or why not? This function is not periodic!!!

Practice ProblemsState the domain and range of the given functions, and sketcha graph showing four periods.

cosy x1.

2 ,2 by 0.25,1.25 Graph window:

, Domain: 0,1Range:

cosy x2.

4 ,4 by 1.2,1.2 Graph window:

, Domain: 1,1Range:

2siny x3.

2 ,2 by 1.25,0.25 Graph window:

, Domain: 1,0Range:

Practice ProblemsThe graphs of the given functions oscillate between two parallellines. Find the equations of the two lines, and graph eachfunction in the same window with its respective lines.

1 0.5 cos 2y x x 1.

10,10 by 10,10 Graph window:

0.5 2 0.5x y x The lines:

1 3cosy x x 2.

10,10 by 10,10 Graph window:

2 4x y x The lines: