starting off the day with a problem… the bolivar lighthouse is located on a small island 350ft...
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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 x
d 350secd x
(b) If x is 1.55 rad, what is d ?
350sec 1.55d 16831.108 ft
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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
ft
800tan y
xP
(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
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GRAPHS OF COMPOSITE
TRIGONOMETRIC FUNCTIONSThe beginning of
Section 4.6a
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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!!!
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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 =
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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
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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:
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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:
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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)
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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 sine
3sin x Changing notation f x
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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)
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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:
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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.
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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!!!
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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:
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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: