Find the value of x. Then find the value of sin θ, cos θ, and tan θ for the triangle. 1.

Transcription

1 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.. March 30, 2017 Geometry 9.6 Solving Right Triangles 1

2 Geometry 9.6 Solving Right Triangles

3 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? March 30, 2017 Geometry 9.6 Solving Right Triangles 3

4 Goals Use inverse trig functions to find angle measures. Solve right triangles. Solve problems using right triangles. March 30, 2017 Geometry 9.6 Solving Right Triangles 4

5 Solving a triangle means Finding the lengths of the three sides. Finding the measure of the three angles. B c a A b C In a right triangle, one angle is always 90, thus we don t need to worry about it. March 30, 2017 Geometry 9.6 Solving Right Triangles 5

6 Our Tools to Solve Triangles: Trig equations Pythagorean Theorem Inverse trig functions A calculator for speed and accuracy March 30, 2017 Geometry 9.6 Solving Right Triangles 6

7 Inverse Trig Function A is an angle opp is a trig ratio hyp Trig Function sin A = opp means: use the sine of an angle hyp to find a trig ratio. Inverse Trig Function sin 1 ( opp ) = A means: use the inverse sine hyp of a trig ratio to find the angle. Example: sin 80 = sin = 80 March 30, 2017 Geometry 9.6 Solving Right Triangles 7

8 Inverse Trig Functions If sin A = x, then sin -1 x = A. If cos A = y, then cos -1 y = A. If tan A = z, then tan -1 z = A. March 30, 2017 Geometry 9.6 Solving Right Triangles 8

9 Example 1 sin A = What is A? A = sin -1 (.766) A 50 March 30, 2017 Geometry 9.6 Solving Right Triangles 9

10 Example 2 cos A = What is A? A = cos -1 (.2079) A 78 March 30, 2017 Geometry 9.6 Solving Right Triangles 10

11 Example 3 tan A = What is A? A = tan -1 (.1051) A 6 March 30, 2017 Geometry 9.6 Solving Right Triangles 11

12 Example 4: Solving a triangle 12 A c First, we will find A. tan A = 7/12 A = tan -1 (7/12) A B March 30, 2017 Geometry 9.6 Solving Right Triangles 12

13 Example 4: Solving a triangle 12 A 30.3 c Now find B. tan B = 12/7 B = tan -1 (12/7) B B March 30, 2017 Geometry 9.6 Solving Right Triangles 13

14 Example 4: Solving a triangle A Or c The acute angles of a right triangle are complementary B B = = 59.7 March 30, 2017 Geometry 9.6 Solving Right Triangles 14

15 Example 4: Solving a triangle 12 A c 59.7 B Find side c. Pythagorean Theorem is best because it doesn t use rounded data March 30, 2017 Geometry 9.6 Solving Right Triangles 15 c c c c 13.9

16 Example 4: Solving a triangle 12 A The triangle is solved. Notice: the measures are all approximate B March 30, 2017 Geometry 9.6 Solving Right Triangles 16

17 You try it. Solve the triangle. 15 A First, find angle A. tan A = 32/15 c A = tan -1 (32/15) A B March 30, 2017 Geometry 9.6 Solving Right Triangles 17

18 A 15 You try it. Solve the triangle Next, find angle B. tan B = 15/32 c B 32 B = tan -1 (15/32) B 25.1 or = 25.1 March 30, 2017 Geometry 9.6 Solving Right Triangles 18

19 You try it. Solve the triangle. Now find side c. c A 64.9 c c c B c 1249 c 35.3 March 30, 2017 Geometry 9.6 Solving Right Triangles 19

20 You try it. Solve the triangle. A The triangle is solved B March 30, 2017 Geometry 9.6 Solving Right Triangles 20

21 Example 5: Solve the triangle. b A Find A first, since it s the complement of the other acute angle. A = = 52 a 38 March 30, 2017 Geometry 9.6 Solving Right Triangles 21

22 Example 5: Solve the triangle. A Now use sine to find a. b a sin sin 52 a a 38 a 13 March 30, 2017 Geometry 9.6 Solving Right Triangles 22

23 Example 5: Solve the triangle. A Now use cosine to find b. b b cos cos52 b b 10.2 March 30, 2017 Geometry 9.6 Solving Right Triangles 23

24 Example 5: Solve the triangle. A The triangle is solved March 30, 2017 Geometry 9.6 Solving Right Triangles 24

25 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!! March 30, 2017 Geometry 9.6 Solving Right Triangles 25

26 Your Turn: Solve this triangle. A 25 c 10 B March 30, 2017 Geometry 9.6 Solving Right Triangles 26

27 Your Turn: Solution A c 2 = c26.9 c 2 = 725 c 26.9 tan B = 25/ B B = tan -1 (25/10) B = 68.2 A = = 21.8 March 30, 2017 Geometry 9.6 Solving Right Triangles 27

28 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. March 30, 2017 Geometry 9.6 Solving Right Triangles 28

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? Jim ft. March 30, 2017 Geometry 9.6 Solving Right Triangles 29

30 Example 6: Solution tan x x 110 tan 7.5 Jim x ft. 7.5 x March 30, 2017 Geometry 9.6 Solving Right Triangles 30

31 Example 6: Solution sin y y 110 sin 7.5 Jim y y ft. March 30, 2017 Geometry 9.6 Solving Right Triangles 31

32 Example 6: Is this correct? YES! Jim ft. March 30, 2017 Geometry 9.6 Solving Right Triangles 32

33 Example 6: Indirect Measure Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly. Jim ft. March 30, 2017 Geometry 9.6 Solving Right Triangles 33

34 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. March 30, 2017 Geometry 9.6 Solving Right Triangles 34

35 Homework March 30, 2017 Geometry 9.6 Solving Right Triangles 35