Applications of Trigonometric Functions in Physics
This activity has two purposes.
First, to help you review trigonometric functions, which will be used extensively in oscillations and
waves.
Second, to help you “read the story” that is hidden in a trig function. The last exercise is an example
of how you can learn a lot about a trig function without ever plugging in any numbers.
Ideally, the idea is to see how much you can know without a calculator. However, because of the
online nature of this course, I suggest you use an online graphing tool like Desmos
(https://www.desmos.com/calculator) to make the requested plots (and you can easily export plot
images via Desmos to include in your submission) and to play around with the parameters of the
relevant trigonometric functions. (Also see https://learn.desmos.com/ for tips on how to use Desmos.)
The Unit Circle:
Sine and cosine also define the x and y coordinate pairs on the unit circle (circle of radius 1) since we can
form a right triangle around the angle as follows:
Thus
sin
opp y
y
hyp 1
cos
adj x
x
hyp 1
1. (5 points) Three important angles in the unit circle are 30o (=/6 rads), 45o (=/4 rads), 60o (=/3
rads). Use the following triangles and definitions of trig functions to find the sines and cosines of these
angles (it’s okay to leave as a fraction with a square root):
1
1/2
cos(30o)=
30o
sin(30o)=
√3/2
cos(60o)=
sin(60o)=
1
cos(45o)=
1/√2
45o
o
sin(45 )=
1/√2
2
Using information on the previous page, the angles provided, and that fact that this is a unit circle, fill in
the blanks below (each set of blanks are the x,y coordinates of the nearest large dot).
y
A
B
C
o
90
60o
45o
D
o
30
180o
G
E
o
270
x
F
3
2. (5 points) Using the unit circle to guide your thinking,
a.
Explain why |cos | and |sin | are always less than or equal to one.
b.
For what range of angles (between 0 and 2) is sine positive? Negative?
c.
For what range of angles (between 0 and 2) is cosine positive? Negative?
Note: When you sketch sine and cosine, the values at =0, /2, , 3/2 and 2 are easy to remember
and can guide you in sketching the whole function. Use your graphing calculator to get a sense of the
function in between these points. We will call these the “anchor points” for the function.
Lastly, we need to define a few terms. Any function that repeats is called a periodic function; each
repetition is called a cycle. Here is an example showing three cycles of a periodic function:
0o
30o
60o
90o
The period of this function is 30o; the period is the shortest interval after which the function repeats.
4
The Functions: Sine and Cosine
3. (10 points) Using the values from the Unit Circle, fill in the following table (notice we’ve switched
from degrees to radians).
0
/2
3 / 2
2
5 / 2
3
7 / 2
4
sin
cos
Using the information from the table, sketch 2 full periods of the graphs of the sine and cosine
functions. Be sure you label 3 tick marks along each axes.
Graph of y sin( )
Graph of y cos( )
Period: ______________________________
Period: ______________________________
Maximum Value:_______________________
Maximum Value:_______________________
Minimum Value:_______________________
Minimum Value:_______________________
x-intercept(s):_________________________
x-intercept(s):_________________________
And this concludes the “pure math” review. If any of the concepts covered so far seem confusing or
unfamiliar to you, you should consider doing some additional review.
5
Physical context
To be concrete, picture a coffee cup sitting on the edge of 15 cm radius turntable in a microwave, and
the microwave is running, so the coffee cup is moving in a circle at a constant rate, going once around in
40 seconds. If you get down to eye level with the microwave, you will see just the x-component of its
motion, and (from our knowledge of the unit circle) x=cos().
4. (5 points) To be sure this makes sense, draw a picture of the turntable from the top and draw x-y
axes. Sketch your eye on the picture so that what you see from eye level is the x coordinate. Pick one
location on the circle and indicate both and x for that location of the cup.
What is the x location of the cup at the following times (you are given that it starts at 15 cm)?
t
0s
x location of cup
15cm
10 s
20s
30s
40s
Need for a more refined model
The mathematical model x=cos() is not quite correct or complete. First, we want to include in this
description how changes in time, and second, the turntable is not a unit circle, but a circle of radius 15
cm. In this activity we will learn how to put these refinements in the mathematical model.
How changes in time.
Recall that for an object moving at a constant rate in a straight line (e.g. a car moving down a straight
road) x=x0+vt, where x0 is the location at t=0.
Analogously, when the cup moves around a circle at a constant rate, =0 + va t. (We will take 0=0
from now on). The notation va stands for angular velocity.
5. (5 points) Recalling linear motion for a moment, if an object moves 30m in 5 seconds, what is v?
v=
Using the definition that va = change in angle/change in time, what is va for the cup on the turntable? Be
sure to use radians (not degrees) for the angle. (We use radians to be consistent with the text.)
va =
6
Using va, what angle has the cup traveled 20 seconds? Show your work for full credit and be sure to
keep units. As a check, what fraction of a cycle is 20 seconds? Does that agree with your angle?
s=
fraction of a cycle=
Using va, what angle has the cup traveled in 800 seconds? Show your work for full credit and be sure to
keep units. As a check, how many cycles is 800 seconds? Does that agree with your angle?
800s=
number of cycles=
Do the last two equations have correct units for an angle? If not, look doe the mistake, or check with
the instructor.
6. (5 points) Let’s do one more specific example, but this time assume it takes 12 seconds to go once
around the turntable
What is va= in this example?
v a=
What angle has the cup traveled in 3 seconds? Show your work for full credit and be sure to keep units.
What fraction of a cycle is 3 seconds?
=
fraction of a cycle=
What angle has the cup traveled 48 seconds? Show your work for full credit and be sure to keep units.
How many cycles is 48 seconds?
=
number of cycles=
7
7. (5 points) We end this task with the angular velocity for a motion with an unspecified period T
seconds.
What is the va in this case?
v a=
Using va, what angle has the cup traveled in (T/4) seconds?
=
Using va, what angle has the cup traveled 3T seconds?
=
To check that you have calculated the correct angles, make a sketch of the turntable and make and label
dots where the cup is at T/4 and 3T seconds (using the definition of T). Do these locations agree with
the angles you calculated? If not, check your work and/or get help from the TA.
Lastly, using angular velocity, what angle corresponds to t seconds, where t is an unspecified time?
(Your answer will have both T and t – these are not the same*) (This is boxed because this general
formula will be useful many times as you learn about oscillations and waves.)
t=
Note that the book uses for angular velocity va. This symbol is the Greek letter omega, and looks
just like a w, except with rounded corners.
*
“T” stands for a fixed time to walk around the circle once (but unspecified); “t” stands for the
variable time; think of it as all the various readings of your watch as the cup goes around the
turntable. Sorry to use the same letter (one cap and one lower case), but the book uses the
same convention, so might as well start now!
8
8. (10 points) Now that we’ve seen how depends on time for uniform circular motion, let’s see how
that works in conjunction with the sine and cosine function. In the table below, “s” stands for seconds.
What is the period of the motion, and how can you tell based on the equations for x(t) in the table?
By filling out the “fraction of a cycle” row first, you should be able to fill out the other two rows.
t
0s
1.25 s
2.5s
3.75s
5s
6.25s
7.5s
8.75s
10s
Fraction of a
cycle
æ 2 t ö
x(t) sin ç
÷
è (5s) ø
æ 2 t ö
x(t) cos ç
÷
è (5s) ø
9
9. (10 points) Here are two slightly different trig functions.
a. What is the period of the motion described by the following equation, and how can you tell (be
careful! – one small thing has changed!)
æ t ö
x(t) sin ç
÷
è (5s) ø
period=T=
Reasoning:
b. Now you are given the following equation and asked for the period. Before deciding, read the
following discussion between students
æ t ö
x(t) sin ç
÷
è (7s) ø
Claire: It takes 2 to complete one rotation, and you can see the 2 is missing from the equation, so you
multiply top and bottom by 2 to get it in the right form, giving a period of T=14s.
Betsy: You forgot the , so we have to multiply top and bottom by 2 and the period is 14 s.
Claire: Can the period be 14 s? That’s weird!
Period = T=
Reasoning:
10
What about the 15 cm radius?
10. (20 points) Lastly, we need to take into account that the cup travels on a 15 cm radius turntable,
and not the unit circle. In this problem you will decide which of the following mathematical models
describes that motion. Only one is correct. First, look back at page 6, where you wrote down the values
for the cup location; your math model should agree with these values. Next, decide if each model is
correct or incorrect. Explain your answer based on a plot of the function, looking at min and max values
of the function, evaluating the function at t=0 (you need NOT evaluate at all the times unless you want
verification), OR an argument based on units.
t
0s
10 s
20s
30s
40s
0s
10 s
20s
30s
40s
0s
10 s
20s
30s
40s
0s
10 s
20s
30s
40s
Fraction of a cycle
æ 2 t ö
cos ç
÷ +15cm
è (40s) ø
t
Fraction of a cycle
æ 2 t
ö
cos ç
+15cm ÷
è (40s)
ø
t
Fraction of a cycle
æ 2 t ö
15cm * cos ç
÷
è (40s) ø
t
Fraction of a cycle
æ 2 t
ö
cos ç
*15cm ÷
è (40s)
ø
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11. (20 points) As a check on your understanding, sketch the graphs of the following functions. (Include
two full periods and label 3 tick marks along each axis.) (In the equations below “m” is for meters and
“s” is for seconds.) Note that these are NOT the position and velocity for the same object.
æ2 t ö
÷
è(3s) ø
position= x (5m)sin ç
æ2 t ö
÷
è(3s) ø
Graph of x (5m)sin ç
Period: ______________________________
Maximum Value:_______________________
Minimum Value:_______________________
x-intercept(s):__________________________
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