1. Geometry 9.6 Solving Right Triangles
9.6 Warmup
Find the value of x. Then find the value of sin θ , cos θ , and tan θ for the triangle. 1. Find the value of the unknown sides. 2. .
November 13, 2018
Geometry 9.6 Solving Right Triangles

2. 9.6 Solving Right Triangles
Geometry
9.6 Solving Right Triangles

3. Geometry 9.6 Solving Right Triangles
9.6 Essential Question
When you know the lengths of the sides of a right triangle, how can you find the measures of the two acute angles?
November 13, 2018
Geometry 9.6 Solving Right Triangles

4. Geometry 9.6 Solving Right Triangles
Goals
Use inverse trig functions to find angle measures.
Solve right triangles.
Solve problems using right triangles.
November 13, 2018
Geometry 9.6 Solving Right Triangles

5. 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, thus we don’t need to worry about it. A c b B a C
November 13, 2018
Geometry 9.6 Solving Right Triangles

6. Our Tools to Solve Triangles:
Trig equations
Pythagorean Theorem
Inverse trig functions
A calculator – for speed and accuracy
November 13, 2018
Geometry 9.6 Solving Right Triangles

7. Inverse Trig Function 𝑨 𝑖𝑠 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑜𝑝𝑝 ℎ𝑦𝑝 𝑖𝑠 𝑎 𝑡𝑟𝑖𝑔 𝑟𝑎𝑡𝑖𝑜
𝑨 𝑖𝑠 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑜𝑝𝑝 ℎ𝑦𝑝 𝑖𝑠 𝑎 𝑡𝑟𝑖𝑔 𝑟𝑎𝑡𝑖𝑜
Trig Function
sin 𝐴= 𝑜𝑝𝑝 ℎ𝑦𝑝 ⁡means: use the sine of an angle to find a trig ratio.
Inverse Trig Function
sin −1 ( 𝑜𝑝𝑝 ℎ𝑦𝑝 )=𝐴 means: use the inverse sine of a trig ratio to find the angle.
Example:
sin 80 =
sin = 80
November 13, 2018
Geometry 9.6 Solving Right Triangles

8. 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.
November 13, 2018
Geometry 9.6 Solving Right Triangles

9. Geometry 9.6 Solving Right Triangles
Example 1
sin A = What is A?
A = sin-1(.766)
A  50
November 13, 2018
Geometry 9.6 Solving Right Triangles

10. Geometry 9.6 Solving Right Triangles
Example 2
cos A = What is A?
A = cos-1(.2079)
A  78
November 13, 2018
Geometry 9.6 Solving Right Triangles

11. Geometry 9.6 Solving Right Triangles
Example 3
tan A = What is A?
A = tan-1(.1051)
A  6
November 13, 2018
Geometry 9.6 Solving Right Triangles

12. Example 4: Solving a triangle
First, we will find A.
tan A = 7/12
A = tan-1(7/12)
A  30.3 A c 12 B 7
November 13, 2018
Geometry 9.6 Solving Right Triangles

13. Example 4: Solving a triangle
Now find B.
tan B = 12/7
B = tan-1(12/7)
B  59.7 A
30.3 c 12 B 7
November 13, 2018
Geometry 9.6 Solving Right Triangles

14. Example 4: Solving a triangle
Or…
The acute angles of a right triangle are complementary.
B = 90 – 30.3 = 59.7 A
30.3 c 12
59.7 B 7
November 13, 2018
Geometry 9.6 Solving Right Triangles

15. Example 4: Solving a triangle
Find side c.
Pythagorean Theorem is best because it doesn’t use rounded data. A
30.3 c 12
59.7 B 7
November 13, 2018
Geometry 9.6 Solving Right Triangles

16. Example 4: Solving a triangle
The triangle is solved.
Notice: the measures are all approximate. A
30.3
13.9 12
59.7 B 7
November 13, 2018
Geometry 9.6 Solving Right Triangles

17. You try it. Solve the triangle.
First, find angle A.
tan A = 32/15
A = tan-1(32/15)
A  64.9 A c 15 B 32
November 13, 2018
Geometry 9.6 Solving Right Triangles

18. You try it. Solve the triangle.
Next, find angle B.
tan B = 15/32
B = tan-1(15/32)
B  25.1
or…
90 – 64.9 = 25.1 A c
64.9 15 B 32
November 13, 2018
Geometry 9.6 Solving Right Triangles

19. You try it. Solve the triangle.
Now find side c. A c
64.9 15
25.1 B 32
November 13, 2018
Geometry 9.6 Solving Right Triangles

20. You try it. Solve the triangle.
The triangle is solved. A
35.3
64.9 15
25.1 B 32
November 13, 2018
Geometry 9.6 Solving Right Triangles

21. Example 5: 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
November 13, 2018
Geometry 9.6 Solving Right Triangles

22. Example 5: Solve the triangle.
Now use sine to find a.
52
16.5 b
38 a
November 13, 2018
Geometry 9.6 Solving Right Triangles

23. Example 5: Solve the triangle.
Now use cosine to find b.
52
16.5 b
38
13.0
November 13, 2018
Geometry 9.6 Solving Right Triangles

24. Example 5: Solve the triangle.
The triangle is solved.
52
16.5
10.2
38
13.0
November 13, 2018
Geometry 9.6 Solving Right Triangles

25. 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 to not use rounded data in any calculation.
Be very careful using a calculator. CHECK EVERYTHING TWICE!!
November 13, 2018
Geometry 9.6 Solving Right Triangles

26. Your Turn: Solve this triangle.c 25 B 10
November 13, 2018
Geometry 9.6 Solving Right Triangles

27. Geometry 9.6 Solving Right Triangles
Your Turn: Solution A
c2 =
c2 = 725
c  26.9
tan B = 25/10
B = tan-1(25/10)
B = 68.2
A = 90 – 68.2 = 21.8 c
26.9 25
21.8
68.2 B 10
November 13, 2018
Geometry 9.6 Solving Right Triangles

28. 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.
It’s a fundamental process used in surveying, map making, astronomy and other applications.
November 13, 2018
Geometry 9.6 Solving Right Triangles

29. Example 6: 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 7.5. How far is he from each object?
110 ft.
Jim
7.5
November 13, 2018
Geometry 9.6 Solving Right Triangles

30. Geometry 9.6 Solving Right Triangles
Example 6: Solution
110 ft.
Jim
7.5 x
November 13, 2018
Geometry 9.6 Solving Right Triangles

31. Geometry 9.6 Solving Right Triangles
Example 6: Solution y
110 ft.
Jim
7.5
835.5
November 13, 2018
Geometry 9.6 Solving Right Triangles

32. Example 6: Is this correct?
YES!
842.7
110 ft.
Jim
7.5
835.5
November 13, 2018
Geometry 9.6 Solving Right Triangles

33. Example 6: Indirect Measure
Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly.
842.7
110 ft.
Jim
7.5
835.5
November 13, 2018
Geometry 9.6 Solving Right Triangles

34. 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 the given data to calculate values, when possible.
November 13, 2018
Geometry 9.6 Solving Right Triangles

35. Geometry 9.6 Solving Right Triangles
Homework
November 13, 2018
Geometry 9.6 Solving Right Triangles