1. Geometry 9.6 Solving Right Triangles

2. July 2, 2015Geometry 9.6 Solving Right Triangles2 Goals Use inverse trig functions to find angle measures. Solve right triangles. Solve problems using right triangles.

3. Inverse functions in trig holt homework help 8.2 July 2, 2015Geometry 9.6 Solving Right Triangles3

4. July 2, 2015Geometry 9.6 Solving Right Triangles4 Solving a triangle means… Finding the lengths of the three sides. Finding the measure of the three angles. A BC b a c In a right triangle, one angle is always 90  and we don’t need to worry about it.

5. July 2, 2015Geometry 9.6 Solving Right Triangles5 We can use… Trig equations Pythagorean Theorem Inverse trig functions

6. July 2, 2015Geometry 9.6 Solving Right Triangles6 Inverse Trig Functions If sin A = x, then sin -1 x = A. If cos A = x, then cos -1 x = A. If tan A = x, then tan -1 x = A.

7. July 2, 2015Geometry 9.6 Solving Right Triangles7 Example 1 Sin A = 0.7660. What is A? Sin -1 (.766) = A Use 2 nd sin (.766) in your calculator ***MAKE SURE YOU ARE IN DEGREE MODE******** A  50 

8. July 2, 2015Geometry 9.6 Solving Right Triangles8 Example 2 Cos A = 0.2079. What is A? Cos -1 (.2079) = A A  78 

9. July 2, 2015Geometry 9.6 Solving Right Triangles9 Example 3 Tan A = 0.1051. What is A? Tan -1 (.1051) = A A  6 

10. July 2, 2015Geometry 9.6 Solving Right Triangles10 Solving a triangle 12 7 A B c First, we will find  A. tan A = 7/12 tan -1 (7/12) = A A  30 

11. July 2, 2015Geometry 9.6 Solving Right Triangles11 Solving a triangle 12 7 A B c Now find  B. Since A and B are complementary,  B is about 60 . 30 

12. July 2, 2015Geometry 9.6 Solving Right Triangles12 Solving a triangle 12 7 A B c Find side c. Pythagorean Theorem 30  60 

13. July 2, 2015Geometry 9.6 Solving Right Triangles13 Solving a triangle 12 7 A B 13.9 The triangle is solved. Notice: the measures are all approximate. 30  60 

14. July 2, 2015Geometry 9.6 Solving Right Triangles14 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 

15. July 2, 2015Geometry 9.6 Solving Right Triangles15 You try it. Solve the triangle. 15 32 A B c Next, find angle B. 90 – 65 = 25  65 

16. July 2, 2015Geometry 9.6 Solving Right Triangles16 You try it. Solve the triangle. 15 32 A B c Now find side c. 65  25 

17. July 2, 2015Geometry 9.6 Solving Right Triangles17 You try it. Solve the triangle. 15 32 A B 35.3 The triangle is solved. 65  25 

18. July 2, 2015Geometry 9.6 Solving Right Triangles18 ExampleSolve 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 

19. July 2, 2015Geometry 9.6 Solving Right Triangles19 ExampleSolve the triangle. 16.5 38  A b a Now use sine to find a. 52 

20. July 2, 2015Geometry 9.6 Solving Right Triangles20 ExampleSolve the triangle. 16.5 38  A b 13.0 Now use cosine to find b. 52 

21. July 2, 2015Geometry 9.6 Solving Right Triangles21 ExampleSolve the triangle. 16.5 38  A 10.2 13.0 The triangle is solved. 52 

22. 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.

23. July 2, 2015Geometry 9.6 Solving Right Triangles23 Solve this triangle. 25 10 A c B

24. July 2, 2015Geometry 9.6 Solving Right Triangles24 Solution 25 10 A c B c 2 = 25 2 + 10 2 c 2 = 725 c  26.9 tan B = 25/10 B = tan -1 (2.5) B ≈ 68  A = 90 – 68 = 22  26.9 68  22 

25. July 2, 2015Geometry 9.6 Solving Right Triangles25 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.

26. July 2, 2015Geometry 9.6 Solving Right Triangles26 ProblemUsing a transit. Jim 110 ft. 7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?

27. July 2, 2015Geometry 9.6 Solving Right Triangles27 ProblemSolution Jim 110 ft. 77 x

28. July 2, 2015Geometry 9.6 Solving Right Triangles28 ProblemSolution Jim 110 ft. 77 896 y

29. July 2, 2015Geometry 9.6 Solving Right Triangles29 Is this correct? Jim 110 ft. 77 896 902 YES!

30. July 2, 2015Geometry 9.6 Solving Right Triangles30 Indirect Measure Jim 110 ft. 77 896 902 Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly.

31. July 2, 2015Geometry 9.6 Solving Right Triangles31 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.