T.5
Trigonometry Review
Many of the basic applications of physics, both to mechanical systems and to the
properties of the human body, require a thorough knowledge of the basic properties of
right triangles, which are triangles that have one angle that is equal to 90o.
Furthermore, many of the problems of rotational motion depend upon the radian
measurement of angles.
Basic Trigonometric Relationships
The following trigonometric functions are those that would be used most frequently
in physics applications. Refer to the figure below for identification of the terms in each
expression.
: angle
a: side adjacent the angle
o: side opposite the angle
h: hypotenuse of the right triangle
o
sin
h
a
cos
h
o sin
tan
a cos
a2 o2 h2
Do the Self-Check to see if you need to refresh your knowledge of the use of the
trigonometry of the right triangle. The answers to the Self-Check are found at the end of
this section.
Self- Check
Use the right triangle shown below to answer questions 1 - 5.
1. sin _______
2. cos _______
3. tan _______
4. _______
5. _______
6. A certain right triangle has two sides of lengths a
= 14 cm and b = 20 cm. What is the length of the hypotenuse?
Use the right triangle shown below to answer questions 7 and 8.
7. a = ________
8. b = ________
9. A surveyor wishes to determine the distance between two points A and B, but
cannot make a direct measurement because a river intervenes. The surveyor
steps off a line AC at 90o to AB and 264 meters long. With the transit at point C,
the surveyor measures the angle between line AB and the line formed by C and B.
Angle BCA is measured to be 62o. What is the distance from A to B?
Trigonometry Review
Trigonometry is used extensively in physics. Any time a direction must be
indicated for a quantity, there is a very good possibility that trigonometry will be
required. In many physics applications, knowledge of the properties of right triangles is
sufficient for the description of the situation. Right angle trigonometry is based on
certain ratios and relationships that hold for all right triangles.
Properties of Right Triangles
The basic trigonometric functions of the angles of a right triangle are defined in
terms of ratios of the sides of the triangle. Angles are measured either in units of
degrees or radians.
If two lines intersect, the angle between the lines is defined by drawing a circle
around the intersecting lines, with the point of intersection at the center of the circle.
The complete circle is defined to have a measure of 360o; the angle measure in
degrees for the intersecting lines is then the fraction of the 360 o enclosed by the lines.
For example, if an angle defined by 2 intersecting lines encloses 1/6 of the circle, the
angle measure is (1/6)3600 or 60o.
The radian measure is defined in a similar manner. The circle is defined to have a
measure of 2 radians. An angle defined by two intersecting lines that enclose 1/6 of
the circle would have a measure of (1/6)(2 radians) = /3 radians. The relationship
between degree measure and radian measure can be seen in the figure below.
Since a full circle has a measure of 360o, or alternately, 2 radians, these
measures can be used to construct a conversion factor between the two measures.
The conversion factor can then be used to convert any degree measure to radians or
vice versa.
2 rad 360o
2 rad rad 2
1 Converting 120o to radians: 120o rad 2.1 rad
180 3
o o
360
rad
1
180o
The length of the arc contained by the two intersecting lines is proportional to the
angle between the two lines. If the angle is given in radian measure, this proportionality
can be expressed as s r , where r is the radius of the circle. This relationship shows
s
that the angle in radians is the ratio between the arc length s and the angle : .
r
Since the radian is defined as a ratio between two lengths, it is dimensionless; that is, it
has no units associated with it.
The trigonometric functions are also defined in terms of ratios. Shown below are
three similar triangles, all of which have the same angles. Use the figure and the given
information and complete the calculations necessary to fill in the second table below.
= 53o = 37o
Triangle Side a Side b Side c
A 22.5 mm 30 mm 37.5 mm
B 30 cm 40 cm 50 cm
C 45 m 60 m 75 m
Triangle A Triangle B Triangle C
+
+ +
right angle
a
c
b
c
a
b
Notice that the angles are the same for all the triangles. Does the difference in the
lengths of the sides make a difference to the value of the angle ? _______________
What determines the difference between angle and angle ?
__________________________________________________ Look at the sum of
angles and for each triangle. What do you notice?______________ From this
result, what general statement can be made about the sum of the acute angles in a right
triangle? _____________________ A pair of angles such as these, that add up to 90 o,
are known as complimentary angles. What general statement can be made about the
sum of all of the angles in a triangle? _____________________
Take a look at the ratio of b over c. For all three of the triangles, this is the ratio
of the side opposite the angle to the hypotenuse. What do you notice about these
ratios? ___________________________________ The ratio of the side opposite an
angle in a triangle to the hypotenuse of that triangle is defined to be the sine of the
angle, indicated by sin . Using your calculator, find the sine of the angle . _______
Does this verify your result from the table?___________ Look at the figure and
determine the value of sin . ___________ Verify your result using your calculator.
Take a look at the ratio of a over c. For all three of the triangles, this is the ratio
of the side adjacent the angle to the hypotenuse. What do you notice about these
ratios? ___________________________________ The ratio of the side adjacent an
angle in a triangle to the hypotenuse of that triangle is defined to be the cosine of the
angle, indicated by cos . Using your calculator, find the cosine of the angle .
___________ Does this verify your result from the table? ____________ Look at the
figure and determine the value of cos . __________ Verify your result using your
calculator.
Take a look at the ratio of a over b. For all three of the triangles, this is the ratio
of the side adjacent the angle to the side opposite the angle . What do you notice
about these ratios? ___________________________________ The ratio of the side
adjacent an angle in a triangle to the side opposite the angle of that triangle is defined
to be the tangent of the angle, indicated by tan . Using your calculator, find the
tangent of the angle . _______ Does this verify your result from the table?
____________ Look at the figure and determine the value of tan . ___________
Verify your result using your calculator. Take the ratio of sin to cos . Compare this
to your answer for tan . What did you find? __________________________
In the diagram shown, angles and are complimentary, and angles and
are complimentary. Using what you have learned about the basic trig functions, use the
diagram to fill in the table below.
Angle Sin (Angle) Cos (Angle)
Look at the results in your table. Can you express a general statement about
complimentary angles and their trigonometric functions?_____________
_______________________________________________________________ If you
are having trouble with this look, for example, at the cos and the sin . What do you
notice? ______________________________________________
For each triangle shown in the previous figure, square the two legs (not the
hypotenuse) of the triangle and add them together. For Triangle D: _______ For
Triangle E:_________ Now square the hypotenuse of each triangle and compare that
result to the sum for the corresponding triangle. For Triangle D:________ For Triangle
E: _________ What relationship is there between the square of the hypotenuse of a
right triangle and the sum of the squares of its two legs?
__________________________________ This relationship is known as the
Pythagorean theorem, and it is valid for all right triangles.
Inverse Trigonometric Functions
The trigonometric functions can also be used to find unknown angles if the
lengths of the sides of the triangle are known. Consider again triangles D and E.
From the calculations already done, we know that the sin = 4/5 = 0.8. But how
do we find ? What we are really asking here is “What angle has a sine which is 0.8?”
This question is answered by the inverse trigonometric functions. The inverse sine,
written sin-1, gives a result that is the angle which has the sine given by the argument of
the sin-1. For example, sin-10.8 will give as a result the angle that has a sine of 0.8.
Likewise, cos-10.8 would give as a result the angle that has a cosine of 0.8, and the tan -
1
0.8 would give the angle whose tangent is 0.8. For most calculators, the inverse
trigonometric functions are obtained by first pushing an “inverse” or “2nd” button,
followed by the desired trigonometric function. If you are unsure of how to obtain the
inverse trig functions on your particular calculator, ask your instructor. Use your
calculator and your results for Triangles D and E to find the unknown angles. =
_________ = __________ = __________ = ____________
Practice Problems in Trigonometry
Convert the following angles to radian measures, and find their sine, cosine and tangent
values using your calculator.
1. 60o
2. 53o
3. 37o
4. Use the figure shown to fill in the following table.
Triangle A b
A
B
5. One acute angle of a right triangle is 20o. The length of the hypotenuse is 6 cm.
Find the lengths of the other two sides of the triangle.
6. A car is traveling northeast (45o N of E). If the car traveled a total distance of
42.42 km, how far north did it travel? How far east?
7. Melissa left her house and jogged a distance of 12 blocks due east, then 15 blocks
due north before stopping at her favorite snack bar for a fruit smoothie. How far, in
a straight line, was she from her house when she stopped? What angle does that
straight-line distance make from due east?
8. To find the height of a tall building, a physics student steps 75 paces (each 1
meter) from the base of the building. Using a ruler at arm's length (1 meter), the
student finds that at this distance, the building appears to be 50 centimeters high
as compared to the ruler. Determine the approximate height of the building.
Answers to Self- Check
15 3
1. sin 0.6
25 5
20 4
2. cos 0.8
25 5
20 4
3. tan 1.33
15 3
4. = 37o
5. = 53o
6. 596 cm 24 .4 cm
7. a = 34.5 cm
8. b = 28.9 cm
9. AB = tan 62o = 496.5 m
Answers to Practice Problems
1. = 1.05 rad; sin = 0.866; cos = 0.5; tan = 1.732
2. = 0.925 rad; sin = 0.799; cos = 0.602; tan = 1.327
3. = 0.646 rad; sin = 0.602; cos = 0.799; tan = 0.754
4. Triangle A: a = 1.91, b = 0.58, = 17o
Triangle B: a = 4.53, b = 3.94, = 41o
5. a = 6 cos 20o = 5.64 cm; o = 6 sin 20o = 2.05 cm
6. N = 42.42 sin 45o = 30 m; E = 42.42 cos 45o = 30 m
7. d = 19.2 blocks; = 51.3o 8. h = 37.5 m