1. 9.6 Solving Right Triangles
Geometry
EQ: How do you use trig ratios to find missing acute angles of a right triangles when you know the side lengths?

2. Geometry 9.6 Solving Right Triangles
Goals
Use inverse trig functions to find angle measures.
Solve right triangles.
Solve problems using right triangles.
May 16, 2018
Geometry 9.6 Solving Right Triangles

3. Solving a triangle means…
Finding the lengths of the three sides.
Finding the measure of the three angles.
In a right triangle, one angle is always 90 and we don’t need to worry about it. A c b B a C
May 16, 2018
Geometry 9.6 Solving Right Triangles

4. Geometry 9.6 Solving Right Triangles
We can use…
Trig equations
Pythagorean Theorem
Inverse trig functions
May 16, 2018
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.
May 16, 2018
Geometry 9.6 Solving Right Triangles

6. Geometry 9.6 Solving Right Triangles
Example 1
Sin A = What is A?
Sin-1(.766) = A
Look at your trig table
Under the column for Sine, find the closest value to .766
A  50
Without a calculator, getting any more accurate results is exceedingly difficult.
May 16, 2018
Geometry 9.6 Solving Right Triangles

7. Geometry 9.6 Solving Right Triangles
Example 2
Use a trig table or inverse function on calculator.
Cos A = What is A?
Cos-1(.2079) = A
A  78
Be careful, remember angle measures wont be negative and the ratio to side lengths and angle measures need to make sense.
Its easy to make errors working too fast or carelessly on calculators.
May 16, 2018
Geometry 9.6 Solving Right Triangles

8. Geometry 9.6 Solving Right Triangles
Example 3
Tan A = What is A?
Tan-1(.1051) = A
A  6
May 16, 2018
Geometry 9.6 Solving Right Triangles

9. Geometry 9.6 Solving Right Triangles
Solving a triangle
First, we will find A.
tan A = 7/12
tan A  .5833
tan-1(.5833) = A
The closest we find is under tangent.
A  30 A c 12 B 7
May 16, 2018
Geometry 9.6 Solving Right Triangles

10. Geometry 9.6 Solving Right Triangles
Solving a triangle
Now find B.
Since A and B are complementary, B is about 60. A
30 c 12 B 7
May 16, 2018
Geometry 9.6 Solving Right Triangles

11. Geometry 9.6 Solving Right Triangles
Solving a triangle
Find side c.
Pythagorean Theorem A
30 c 12
60 B 7
May 16, 2018
Geometry 9.6 Solving Right Triangles

12. Geometry 9.6 Solving Right Triangles
Solving a triangle
The triangle is solved.
Notice: the measures are all approximate. A
30
13.9 12
60 B 7
May 16, 2018
Geometry 9.6 Solving Right Triangles

13. You try it. Solve the triangle.
First, find angle A.
tan A = 32/15
tan A 
tan-1(2.1333) = A
A  65 A c 15 B 32
May 16, 2018
Geometry 9.6 Solving Right Triangles

14. You try it. Solve the triangle.
Next, find angle B.
90 – 65 = 25 A
65 c 15 B 32
May 16, 2018
Geometry 9.6 Solving Right Triangles

15. You try it. Solve the triangle.
Now find side c. A
65 c 15
25 B 32
May 16, 2018
Geometry 9.6 Solving Right Triangles

16. You try it. Solve the triangle.
The triangle is solved. A
65
35.3 15
25 B 32
May 16, 2018
Geometry 9.6 Solving Right Triangles

17. Example Solve the triangle.
Find A first, since it’s the complement of the other acute angle.
A = 90 – 38 = 52
52
16.5 b
38 a
May 16, 2018
Geometry 9.6 Solving Right Triangles

18. Example Solve the triangle.
Now use sine to find a.
52
16.5 b
38 a
May 16, 2018
Geometry 9.6 Solving Right Triangles

19. Example Solve the triangle.
Now use cosine to find b.
52
16.5 b
38
13.0
May 16, 2018
Geometry 9.6 Solving Right Triangles

20. Example Solve the triangle.
The triangle is solved.
52
16.5
10.2
38
13.0
May 16, 2018
Geometry 9.6 Solving Right Triangles

21. Geometry 9.6 Solving Right Triangles
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. Check everything twice.
May 16, 2018
Geometry 9.6 Solving Right Triangles

22. Geometry 9.6 Solving Right Triangles
Solve this triangle. A c 25 B 10
May 16, 2018
Geometry 9.6 Solving Right Triangles

23. Geometry 9.6 Solving Right Triangles
Solution A
c2 =
c2 = 725
c  26.9
tan B = 25/10
B = tan-1(2.5)
B = 68
A = 90 – 68 = 22 c
26.9 25
22
68 B 10
May 16, 2018
Geometry 9.6 Solving Right Triangles

24. Geometry 9.6 Solving Right Triangles
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.
May 16, 2018
Geometry 9.6 Solving Right Triangles

25. 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
May 16, 2018
Geometry 9.6 Solving Right Triangles

26. Geometry 9.6 Solving Right Triangles
Problem Solution
110 ft.
Jim 7 x
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Geometry 9.6 Solving Right Triangles

27. Geometry 9.6 Solving Right Triangles
Problem Solution y
110 ft.
Jim 7
896
May 16, 2018
Geometry 9.6 Solving Right Triangles

28. Geometry 9.6 Solving Right Triangles
Is this correct?
YES!
902
110 ft.
Jim 7
896
May 16, 2018
Geometry 9.6 Solving Right Triangles

29. Geometry 9.6 Solving Right Triangles
Indirect Measure
Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly.
902
110 ft.
Jim 7
896
May 16, 2018
Geometry 9.6 Solving Right Triangles

30. Geometry 9.6 Solving Right Triangles
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.
May 16, 2018
Geometry 9.6 Solving Right Triangles