Download Solving Right Triangles

Solving Right Triangles
In chapter 7, we defined the trigonometric functions in terms of
coordinates of points on a circle. Now, our emphasis shifts from circles
to triangles.
When certain parts (sides and angles) of a triangle are known, you will
see that trigonometric relationships can be used to find the unknown
parts.
This is called solving a triangle.
For example, if you know the lengths of the sides of a triangle, then you
can find the measures of its angles.
In this section, we will consider how trigonometry can be applied to
right triangles.

cos
sin 
hypotenuse

adjacent
opposite
Example 1. For the right triangle ABC shown, find the value of b to three
significant digits.
B
a  40
c
A
28
b
C
Which trig ratio should we use to find b?
40
opposite 40

b

tan 28 

 75.2
 b tan 28  40
adjacent b
tan 28
How could we find c?
 402  75.22  c 2  c 2  7, 255.04  c  7255.04  85.2
a 2  b2  c 2
How could we find B?
m A  m B  m C  180  90  28  m B  180  m B  62
Example 2. The safety instructions for a 20 ft. ladder indicate that the
ladder should not be inclined at more than a 70º angle with the ground.
Suppose the ladder is leaned against a house at this angle.
Find (a) the distance x from the base of the house to the foot of the ladder
and (b) the height y reached by the ladder.
cos 70 
x
20
 20cos70  x  6.84
The foot of the ladder is about 6.84 ft. from the
base of the house.
sin 70 
y
20
 20sin 70  y
 18.8
The ladder reaches about 18.8 ft above the ground.
x
Example 3. The highest tower in the world is in Toronto, Canada, and is
553 m high. An observer at point A, 100 m from the center of the
tower’s base, sights the top of the tower. The angle of elevation is A.
Find the measure of this angle to the nearest tenth of a degree.
tan A 
553
 5.53
100
A  Tan1  5.53  79.7
Because we can divide an isosceles triangle into two congruent right
triangles, we can apply trigonometry to isosceles triangles.
Example 4. A triangle has sides of lengths 8, 8, and 4. Find the measures
of the angles of the triangle to the nearest tenth of a degree.
cos D 
F
2
 0.25
8
8
D  Cos 1  0.25  75.5
E  D  75.5
F  180  2  75.5  29.0
D
8
2
2
M
E
true
true
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