Geometry
9.6 Solving Right Triangles
Goals



Use inverse trig functions to find angle
measures.
Solve right triangles.
Solve problems using right triangles.
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Geometry 9.6 Solving Right Triangles
2
Inverse functions in trig

holt homework help 8.2
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Geometry 9.6 Solving Right Triangles
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Solving a triangle means…


Finding the lengths of the three sides.
Finding the measure of the three
angles.
In a right
A
B
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c
b
a
C
Geometry 9.6 Solving Right Triangles
triangle,
one angle
is always
90 and
we don’t
need to
worry
about it.
4
We can use…



Trig equations
Pythagorean Theorem
Inverse trig functions
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Geometry 9.6 Solving Right Triangles
5
Inverse Trig Functions



If sin A = x, then sin-1x = A.
If cos A = x, then cos-1x = A.
If tan A = x, then tan-1x = A.
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Geometry 9.6 Solving Right Triangles
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Example 1





Sin A = 0.7660. What is A?
Sin-1(.766) = A
Use 2ndsin (.766) in your calculator
***MAKE SURE YOU ARE IN DEGREE
MODE********
A  50
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Geometry 9.6 Solving Right Triangles
7
Example 2



Cos A = 0.2079. What is A?
Cos-1(.2079) = A
A  78
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Geometry 9.6 Solving Right Triangles
8
Example 3



Tan A = 0.1051. What is A?
Tan-1(.1051) = A
A  6
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Geometry 9.6 Solving Right Triangles
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Solving a triangle
First, we will find A.
A
tan A = 7/12
tan-1 (7/12) = A
c
12
A  30
7
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B
Geometry 9.6 Solving Right Triangles
10
Solving a triangle
Now find B.
A
30
c
12
7
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Since A and B are
complementary, B is
about 60.
B
Geometry 9.6 Solving Right Triangles
11
Solving a triangle
Find side c.
A
12
Pythagorean Theorem
30
c
c  12  7
2
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2
c  144  49
2
60
7
2
B
c  193
c  13.9
2
Geometry 9.6 Solving Right Triangles
12
Solving a triangle
The triangle is solved.
A
Notice: the measures are
all approximate.
30
13.9
12
60
7
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B
Geometry 9.6 Solving Right Triangles
13
You try it. Solve the triangle.
First, find angle A.
tan A = 32/15
A
c
tan-1(32/15) = A
15
32
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B
A  65
Geometry 9.6 Solving Right Triangles
14
You try it. Solve the triangle.
Next, find angle B.
90 – 65 = 25
A
65
c
15
32
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B
Geometry 9.6 Solving Right Triangles
15
You try it. Solve the triangle.
c  15  32
Now find side c.
2
A
2
2
c  225  1024
2
65
c
c  1249
15
2
25
32
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B
c  1249
c  35.3
Geometry 9.6 Solving Right Triangles
16
You try it. Solve the triangle.
The triangle is solved.
A
65
35.3
15
25
32
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B
Geometry 9.6 Solving Right Triangles
17
Example
Solve the triangle.
A
52
b
16.5
Find A first, since it’s
the complement of the
other acute angle.
A = 90 – 38 = 52
38
a
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Geometry 9.6 Solving Right Triangles
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Example

Solve the triangle.
A
Now use sine to find a.
52
b
16.5
38
a
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Geometry 9.6 Solving Right Triangles
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Example
Solve the triangle.
A
Now use cosine to find b.
52
b
16.5
38
13.0
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Geometry 9.6 Solving Right Triangles
20
Example
Solve the triangle.
A
The triangle is solved.
52
10.2
16.5
38
13.0
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Geometry 9.6 Solving Right Triangles
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Important





You can solve a triangle in any order
you want to, as long you have the data
you need for each step.
It’s best not to use rounded data in any
calculation.
Be very careful using a calculator.
Be sure you are in DEGREE Mode
when using your calculator!
Check everything twice.
22
Solve this triangle.
A
25
c
10
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B
Geometry 9.6 Solving Right Triangles
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Solution
A
c2 = 252 + 102
c2 = 725
25
c  26.9
c26.9
tan B = 25/10
B = tan-1(2.5)
68
10
B
B ≈ 68
A = 90 – 68 = 22
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Geometry 9.6 Solving Right Triangles
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Indirect Measure


One of the most powerful uses of trig
is to measure things that can’t be
measured directly. This is indirect
measure.
Fundamental process used in
surveying, map making, astronomy
and other applications.
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Geometry 9.6 Solving Right Triangles
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Problem
Using a transit.
Jim the Surveyor uses a transit to
measure distances. He knows the
distance between the tree and the fire
hydrant is 110 ft. And to move from one
to the other he swings his transit
through 7. How far is he from each
object?
110 ft.
Jim
7
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Geometry 9.6 Solving Right Triangles
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Problem
Solution
110
tan 7 
x
110
110
x

tan 7 .1228
x  896
110 ft.
Jim
7
x
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Geometry 9.6 Solving Right Triangles
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Problem
Solution
110
sin 7 
y
110
110
y

sin 7 .1219
y  902
y
Jim
110 ft.
7
896
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Geometry 9.6 Solving Right Triangles
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Is this correct? YES!
110 ft.
Jim
7
896
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Geometry 9.6 Solving Right Triangles
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Indirect Measure
Using trig, Jim can
determine the distances to
the tree and the fire
hydrant without measuring
them directly.
110 ft.
Jim
7
896
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Geometry 9.6 Solving Right Triangles
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Summary



Solving a triangle means to find all six
parts: 3 angles, 3 sides.
Use inverse trig function (sin-1, cos-1,
tan-1) to find angles.
Use given values when possible.
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Geometry 9.6 Solving Right Triangles
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