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LESSON 1 • Trigonometric Functions 467
IInvest invest iggationation 22 Measuring Without Measuring
Measuring Without MeasuringThe highest point on the Earth’s surface
is the peak of Mount Everest in the Himalaya mountain range along
the Tibet-Nepal border in Asia. The most recent calculations
indicate that Mount Everest rises 8,872 meters (29,108 feet) above
sea level. As early as 1850, surveyors had estimated the height of
that peak with error of only 0.4%. The first climbers known to
actually reach the summit were Tenzing Norgay and Edmund Hillary in
1953.
In Investigation 1, you explored how the tangent and cosine
functions could be used to calculate a height and distance that
could not be measured directly. As you work on the problems of this
investigation, make note of answers to the following question:
How can trigonometric functions be used to calculate heights
like that of Mount Everest and other distances
that cannot be measured directly?
1 The following sketch shows the start of one surveyor’s attempt
to determine the height of a tall mountain without climbing to the
top herself.
B
C2,700 feetA51˚
a. Use the given information to calculate the lengths of −− AB
and −− BC . (Use the table on page 465 to calculate approximate
values of trigonometric functions as needed.)
b. Suppose that a laser ranging device allowed you to find the
length of −− AB and the angle of elevation ∠BAC, but you could not
measure the length of −− AC . How could you use this information
(instead of the information from the diagram) to calculate the
lengths of −− AC and −− BC ?
Name: _______________________________
Unit 6Lesson 1
Investigation 2
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468 UNIT 7 • Trigonometric Methods
2 The trigonometric functions are often used in problems modeled
with right triangles, as in Problem 1. It is helpful to be able to
use these functions without first placing an acute angle of the
triangle in standard position in a coordinate plane. Examine the
diagram of right △ABC with ∠C a right angle.
a. Explain why the following right triangle definitions of sine,
cosine, and tangent make sense.
tangent of ∠A = tan A = a _ b = length of side opposite ∠A
___ length of side adjacent to ∠A
sine of ∠A = sin A = a _ c = length of side opposite ∠A
___ length of hypotenuse
cosine of ∠A = cos A = b _ c = length of side adjacent to ∠A
___ length of hypotenuse
b. Write expressions for tan B, sin B, and cos B.
3 Chicago’s Bat Column, a sculpture by Claes Oldenburg, is shown
below.
a. About how tall do you think the column is? What visual clues
in the photo did you use to make your estimate?
b. In the diagram at the right, what lengths and angles could
you determine easily by direct measurement (and without using
high-powered equipment)?
A
B
C
D
c. Which trigonometric functions of ∠A involve side −− BC ? Of
these, which also involve a measurable length?
d. Which of the trigonometric functions of ∠B involve side −− BC
and a measurable length? If you know the measure of angle of
elevation ∠A, how can you find the measure of ∠B?
A
B
C
a
b
c
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LESSON 1 • Trigonometric Functions 469
e. To find the height of Bat Column, Krista and D’wan proceeded
as follows. First, Krista chose a spot (point A) 20 meters from the
sculpture (point C). D’wan estimated the angle of elevation at A by
sighting the top of the sculpture along a protractor and using a
weight as shown. He measured ∠A to be 55°. What is the measure of
∠B?
f. They next used the following reasoning to find the height
BC.
We need to find BC. We know that tan 55° = BC _ AC .But, tan 55°
= 1.4281 and AC = 20 m.So, we need to solve 1.4281 = BC _ 20 .If we
multiply both sides of the equation by 20, we get BC = 1.4281 • 20,
or about 29 m.
i. Why did they decide to use the tangent function rather than
the sine function?
ii. How did they know that tan 55° = 1.4281? iii. Check that
each step in their reasoning is correct. iv. How do you think
Krista and D’wan used this information to
calculate the height of Bat Column?
g. Ken said he could find the length AB (the line of sight
distance) by solving cos 55° = AC _ AB .
i. Use Ken’s idea to find the length AB. ii. What is another way
you could find AB by using a different
trigonometric function?
iii. Could you find AB without using a trigonometric function?
Explain your reasoning.
4 As you have seen in Problems 1 and 3, an important part of
solving problems using trigonometric methods is to decide on a
trigonometric function that uses given information. For each right
triangle below, write two equations involving trigonometric
functions of acute angles that include s and the indicated length.
Then rewrite each equation in an equivalent form “s = … .”
a.
C B
A
s7
b. F
DE
8s
c.
s
Q
P
R 15
Angle ofElevation
MeasuredAngle
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470 UNIT 7 • Trigonometric Methods
5 Rather than using tables, today it is much easier to find
values of the trigonometric functions with a calculator.
a. To calculate a trigonometric function value for an angle
measured in degrees, first be sure your calculator is set in degree
mode. Then simply press the keys that correspond to the desired
function. For example, to calculate sin 27.5° on most graphing
calculators, press
SIN 27.5 ) ENTER . Try it. Then calculate cos 27.5° and tan
27.5°. Although your calculator displays 10 digits, you should
report estimates of trigonometric function values to the nearest
ten-thousandth.
b. Use your calculator to find the sine, cosine, and tangent of
66°. Of 54°. Compare these results with those in the table on page
465.
c. Use your calculator to find the sine, cosine, and tangent of
45°. Compare these results with the exact values you found in
Problem 7 of Investigation 1.
6 Each part below gives angle measure and side length
information for right △ABC with ∠C a right angle. For each, sketch
and label the triangle. Then find the lengths of the remaining two
sides and find the measure of the third angle.
a. ∠B = 52°, a = 5 m b. ∠A = 48°, a = 15 mic. ∠A = 31°, b = 8
in. d. ∠A = 70°, c = 14 cm
Summarize the Mathematics
The trigonometric functions sine, cosine, and tangent are useful
in calculating lengths in situations modeled with right triangles.
Refer to the right triangle below in summarizing your thinking
about how to use trigonometric functions in the situations
described.
B
A Cb
ac
a If you knew b and the measure of ∠A, how would you use that
information to find a? How could you find m∠B? How could you find
c?
b If you knew c and the measure of ∠B, how would you use that
information to find a? How could you find m∠A? How could you find
b?
c If you knew a and the measure of ∠A, how would you use that
information to find c?
Be prepared to explain your methods to the entire class.
