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Applications of Right Triangles
• Significant Digits
–
–
Represents the actual measurement.
Most values of trigonometric functions and
virtually all measurements are
approximations.
Copyright © 2007 Pearson Education, Inc.
Slide 8-1
5.7 Solving a Right Triangle Given an
Angle and a Side
Example Solve the right triangle ABC, with A = 34º 30 and
c = 12.7 inches.
Solution
Angle B = 90º – A = 89º 60 – 34º 30
= 55º 30.
a
sin A 
c
sin 34 30 

a

 a  12.7 sin 34 30  7.19 inches
12.7
Use given information to find b.
b

cos 34 30 
 b  12.7 cos 34 30  10.5 inches
12.7

Copyright © 2007 Pearson Education, Inc.
Slide 8-2
5.7 Solving a Right Triangle Given Two
Sides
Example Solve right triangle ABC if a = 29.43 centimeters
and c = 53.58 centimeters.
Solution
Draw a sketch showing the given information.
29.43
sin A 
53.58
Using the inverse sine function
on a calculator, we find A  33.32º.
B = 90º – 33.32º  56.68º
Using the Pythagorean theorem,
b c a
2
2
2
b  53.58  29.43
2
Copyright © 2007 Pearson Education, Inc.
2
2
 b  44.77 centimeter s.
Slide 8-3
5.7 Calculating the Distance to a Star
•
•
•
In 1838, Friedrich Bessel determined the distance to a
star called 61 Cygni using a parallax method that relied
on the measurement of very small angles.
You observe parallax when you ride in a car and see a
nearby object apparently move backward with respect to
more distance objects.
As the Earth revolved around the sun, the observed
parallax of 61 Cygni is   .0000811º.
Copyright © 2007 Pearson Education, Inc.
Slide 8-4
5.7 Calculating the Distance to a Star
Example One of the nearest stars is Alpha
Centauri, which has a parallax of   .000212º.
(a) Calculate the distance to Alpha Centauri if the
Earth-Sun distance is 93,000,000 miles.
(b) A light-year is defined to be the distance that
light travels in 1 year and equals about 5.9
trillion miles. Find the distance to Alpha
Centauri in light-years.
Copyright © 2007 Pearson Education, Inc.
Slide 8-5
5.7 Calculating the Distance to a Star
Solution
(a) Let d be the distance between Earth and Alpha
Centauri. From the figure on slide 8-46,
93,000,000
93,000,000
sin  
or d 
d
sin 
93,000,000
d
 2.51 1013 miles.
sin .000212
2.51 1013
 4.3 light - years.
(b) This distance equals
12
5.9  10
Copyright © 2007 Pearson Education, Inc.
Slide 8-6
5.7 Solving a Problem Involving Angle of
Elevation
•
Angles of Elevation or Depression
Example Francisco needs to know the height of a tree. From
a given point on the ground, he finds that the angle of
elevation to the top of the tree is 36.7º. He then moves back
50 feet. From the second point, the angle of elevation is 22.2º.
Find the height of the tree.
Copyright © 2007 Pearson Education, Inc.
Slide 8-7
5.7 Solving a Problem Involving Angle of
Elevation
Analytic Solution There are two unknowns, the distance x
and h, the height of the tree.
h


In triangle ABC, tan 36.7 
or h  x tan 36.7 .
x
h


tan
22
.
2

or
h

(
50

x
)
tan
22
.
2
.
In triangle BCD,
50 x
Each expression equals h, so the expressions must be equal.
x tan 36.7  (50  x ) tan 22.2



x tan 36.7  50 tan 22.2  x tan 22.2



x tan 36.7  x tan 22.2  50 tan 22.2
50 tan 22.2
x
tan 36.7   tan 22.2

Copyright © 2007 Pearson Education, Inc.

Slide 8-8
5.7 Solving a Problem Involving Angle of
Elevation
We saw above that h = x tan 36.7º. Substituting for x,
50 tan 22.2



h
tan
36
.
7
 45 feet.



 tan 36.7  tan 22.2 
Graphing Calculator Solution Superimpose the figure on
the coordinate axes with D at the origin.
Line DB has m = tan 22.2º with yintercept 0. So the equation of line DB
is y = tan 22.2º x.
Similarly for line AB, using the pointslope form of a line, we get the
equation y = [tan 36.7º](x – 50).
Copyright © 2007 Pearson Education, Inc.
Slide 8-9
5.7 Solving a Problem Involving Angle of
Elevation
Plot the lines DB and AB on the graphing calculator and find
the point of intersection.
Line DB: y = tan 22.2º x
Line AB: y = [tan 36.7º](x – 50).
Rounding the information at the bottom of the screen, we see
that h  45 feet.
Copyright © 2007 Pearson Education, Inc.
Slide 8-10