Solving Right Triangles

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9.1 Solving Right Triangles
Objective To use trigonometry to find unknown
sides or angles of a right triangle.
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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.
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Right-Triangle-Based Definitions of Trigonometric
Functions For any acute angle A in standard
position,
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Example 1 For the right triangle ABC shown,
find the value of b and c.
Solution Which trig ratio should we use to
find b?
How could we find c? (without using Pythagorean
Theorem)
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a)
b)
c)
d)
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Example 2 For the right triangle ABC shown,
find the value of b and , and
.
Solution The value of b can be found by
Pythagorean Theorem
Which is trig ratio related angle A and B?
In a right triangle, if we know one side and one
acute angle, or two sides, we will know this
triangle (via the trigonometric functions).
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a)
b)
c)
d)
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Example 3 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.
The foot of the ladder is about 6.84 ft. from the
base of the house.
The ladder reaches about 18.8 ft above the ground.
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Example 4 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 towers
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.
Solution The angle of elevation project the
height of the tower, 553 m, we have
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Because we can divide an isosceles triangle into
two congruent right triangles, we can apply
trigonometry to isosceles triangles.
Example 5 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.
Solution In right triangle FDM, DM 2, FD
8, therefore,
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true
true
true
a) sin? x/z, cos? y/z,
tan? x/y,
a) csc? z/x, sec? z/y,
cot? y/x,
b) sin? y/z, cos? x/z,
tan? y/x,
b) csc? z/y, sec? z/x,
cot? x/y,
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Example 6 From points A and B, 10 m apart, the
angles of elevation of the top of a tower are 40o
and 54o, respectively, as show at the right.
Find the towers height.
Solution In right triangle DBC, DC h, BC
x
or
In right triangle DAC, AB 10
or
We get a linear system as following
Applying Cramers Rule, we have
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Example 7 Use the diagram at below, express x
and y in terms of ?, ?, and a.
C
D
Solution We label the diagram as ABCD as the
right. Then AB x, AC y. In right triangle
BCD,
A
B
In right triangle ABC,
So,
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Challenge!!!
Example 8 Find the exact value of sin18o.
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Solution We draw an isosceles triangle ABC
with ?A 36o, and ?B ?C 72o.
D
Also, we draw the angle bisector BD.
72o
2
x
36o
Then the ? ABD is an isosceles triangle, and AD
BD.
36o
E
1
1
2
So is the ? BCD is an isosceles triangle, and BD
BC. Let
and
From A draw height AE on BC at E. Then AE bisect
BC, or BE EC 1.
Since ? ABC ? BCD
or
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Challenge!!!
Example 8 Find the exact value of sin18o.
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Solution
D
72o
2
x
Discard the negative root. We have
36o
36o
E
1
1
2
In right ? AEC, ?CAE 18o, so
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Challenge!!!
Example 9 Find the exact value of sin36o.
Solution From the last example, we can find
the length of AE
F
In ? BDC, from B draw height BF on CD at F. Then
? ABF is a right triangle, and ?BAF 36o and
?ABF 54o. Point F bisect CD. CF FD 1/2x.
Since ? BDF ? ACE, then
or
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Challenge!!!
Example 9 Find the exact value of sin36o.
Solution
In the right ? ABF,
F
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Easier!!!
Example 10 Find the exact value of sin54o.
Solution In right ? ABF, ?ABF 54o. In ?
BDC, Point F bisect CD. CF FD 1/2x. So,
F
Then
Therefore
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Easiest !!!
Example 11 Find the exact value of sin72o.
Solution In right ? ACE, ?C 72o. Therefore,
F
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Summary By the co-function properties, we know
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Assignment P. 334 1 7 (odd), 13 21 (odd),
4, 8, 14, 18, 23, 27, 39