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April 19, 2023 Geometry 9.6 Solving Right Triangles 2
Goals
Use inverse trig functions to find angle measures.
Solve right triangles. Solve problems using right triangles.
Inverse functions in trig
holt homework help 8.2
April 19, 2023 Geometry 9.6 Solving Right Triangles 3
April 19, 2023 Geometry 9.6 Solving Right Triangles 4
Solving a triangle means…
Finding the lengths of the three sides. Finding the measure of the three
angles.A
B C
b
a
c
In a right triangle, one angle is always 90 and we don’t need to worry about it.
April 19, 2023 Geometry 9.6 Solving Right Triangles 5
We can use…
Trig equations Pythagorean Theorem Inverse trig functions
April 19, 2023 Geometry 9.6 Solving Right Triangles 6
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.
April 19, 2023 Geometry 9.6 Solving Right Triangles 7
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
April 19, 2023 Geometry 9.6 Solving Right Triangles 8
Example 2
Cos A = 0.2079. What is A? Cos-1(.2079) = A A 78
April 19, 2023 Geometry 9.6 Solving Right Triangles 9
Example 3
Tan A = 0.1051. What is A? Tan-1(.1051) = A A 6
April 19, 2023 Geometry 9.6 Solving Right Triangles 10
Solving a triangle
12
7
A
B
c
First, we will find A.
tan A = 7/12
tan-1 (7/12) = A
A 30
April 19, 2023 Geometry 9.6 Solving Right Triangles 11
Solving a triangle
12
7
A
B
c
Now find B.
Since A and B are complementary, B is about 60.
30
April 19, 2023 Geometry 9.6 Solving Right Triangles 12
Solving a triangle
12
7
A
B
c
Find side c.
Pythagorean Theorem
30
60
2 2 2
2
2
12 7
144 49
193
13.9
c
c
c
c
April 19, 2023 Geometry 9.6 Solving Right Triangles 13
Solving a triangle
12
7
A
B
13.9
The triangle is solved.
Notice: the measures are all approximate.30
60
April 19, 2023 Geometry 9.6 Solving Right Triangles 14
You try it. Solve the triangle.
15
32
A
B
c
First, find angle A. tan A = 32/15
tan-1(32/15) = A
A 65
April 19, 2023 Geometry 9.6 Solving Right Triangles 15
You try it. Solve the triangle.
15
32
A
B
c
Next, find angle B. 90 – 65 = 25
65
April 19, 2023 Geometry 9.6 Solving Right Triangles 16
You try it. Solve the triangle.
15
32
A
B
c
Now find side c.
65
25
2 2 2
2
2
15 32
225 1024
1249
1249
35.3
c
c
c
c
c
April 19, 2023 Geometry 9.6 Solving Right Triangles 17
You try it. Solve the triangle.
15
32
A
B
35.3
The triangle is solved.
65
25
April 19, 2023 Geometry 9.6 Solving Right Triangles 18
Example Solve the triangle.
16.5
38
A
b
a
Find A first, since it’s the complement of the other acute angle.
A = 90 – 38 = 52
52
April 19, 2023 Geometry 9.6 Solving Right Triangles 19
Example Solve the triangle.
16.5
38
A
b
a
Now use sine to find a.
52
April 19, 2023 Geometry 9.6 Solving Right Triangles 20
Example Solve the triangle.
16.5
38
A
b
13.0
Now use cosine to find b.
52
April 19, 2023 Geometry 9.6 Solving Right Triangles 21
Example Solve the triangle.
16.5
38
A
10.2
13.0
The triangle is solved.
52
22
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.
April 19, 2023 Geometry 9.6 Solving Right Triangles 24
Solution
25
10
A
c
B
c2 = 252 + 102
c2 = 725
c 26.9
tan B = 25/10
B = tan-1(2.5)
B ≈ 68
A = 90 – 68 = 22
26.9
68
22
April 19, 2023 Geometry 9.6 Solving Right Triangles 25
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.
April 19, 2023 Geometry 9.6 Solving Right Triangles 26
Problem Using a transit.
Jim110 ft.
7
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?
April 19, 2023 Geometry 9.6 Solving Right Triangles 27
Problem Solution
Jim110 ft.
7x
110tan 7
110 110
tan 7 .1228896
x
x
x
April 19, 2023 Geometry 9.6 Solving Right Triangles 28
Problem Solution
Jim110 ft.
7896
110sin 7
110 110
sin 7 .1219902
y
y
y
y
April 19, 2023 Geometry 9.6 Solving Right Triangles 30
Indirect Measure
Jim110 ft.
7896
902
Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly.