Warm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72

Transcription

1 Warm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72 Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest degree. 2. sin cos tan sin -1 (0.34) 6. cos -1 (0.63) 7. tan -1 (2.75)

2 Example 1: Finding Trigonometric Ratios for Angles Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A. tan 103 B. cos 165 C. sin 93 tan 103! 4.33 cos 165! 0.97 sin 93! 1.00

3 Example 2: Finding Trigonometric Ratios for Angles Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A. tan 103 B. cos 165 C. sin 93 tan 103! 4.33 cos 165! 0.97 sin 93! 1.00

4 More Examples Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. a. tan 175 b. cos 92 c. sin 160 tan 175! 0.09 cos 92! 0.03 sin 160! 0.34

5 Applying the Primary Trigonometric Ratios

6 In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE. Two things MUST BE understood: 1. This is the hypotenuse.. This will ALWAYS be the hypotenuse 2. This is 90! this makes the right triangle a right triangle!. Without it, we can not do this trig! we WILL NOT use it in our calculations because we COULD NOT do calculations without it.

7 Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other angles and the 2 other sides. A adjacent We will refer to the sides in terms of their proximity to the angle hypotenuse If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. opposite

8 opposite If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. hypotenuse adjacent B

9 X Remember we won t use the right angle

10 One more thing! " this is the symbol for an unknown angle measure. It s name is Theta. Don t let it scare you! it s like x except for angle measure! it s a way for us to keep our variables understandable and organized.

11 To Remember our Trigonometric Ratios we can think of the following: SohCahToa

12 Three Types Trigonometric Ratios! There are 3 kinds of trigonometric ratios we will learn.! sine ratio! cosine ratio! tangent ratio

13 Trigonometric Ratios Name say Sine Cosine tangent Abbreviation Abbrev. Sin Cos Tan Ratio of an angle measure Sin" = opposite side hypotenuse cos" = adjacent side hypotenuse tan" =opposite side adjacent side

14 Primary Trigonometric Ratios! Sine, Cosine, and Tangent are the primary trigonometric ratios that are used to solve for finding the unknown side of a right angle triangle! Primary Trigonometric Ratios B SinA = a c CosA = b c TanA = a b opp hyp adj hyp opp adj SOH CAH TOA a Side opposite of angle A C b Hypotenuse c Side adjacent of angle A A

15 Definitions! Angle of Elevation The angle between the horizontal and the line of sight when one is looking up at an object! Angle of Depression the angle between the horizontal and the line of sight when one is looking down at an object

16 Example The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.! Determine the length of m in Triangle MNP. M 225 ft The length of the hypotenuse is given The measure of the acute angle P is given m is adjacent to angle P What Trigonometric ratio will we use? Cosine N m 60 P

17 N M 225 ft 60 P m Solution! Using the Cosine ratio CosP = NP PM = m n m Cos 60 = 225 Cos 60! 225 = m m =112.5! Therefore, the length of m is about feet

18 Example! Determine the height of the Eiffel Tower if one is standing 68 m from the base and the angle of elevation to the top is 78 degrees.

19 Solution Top of Tower Height TanP = Opp Adj Tan78 = Opp 68 68(Tan78) = Opp Base Person = Opp Therefore, the height of the Eiffel Tower is m

20 Inverse Trigonometric Ratios! The inverse of Sine, Cosine, and Tangent are used to solve for the unknown angle of elevation or depression of a right angle triangle! Inverse Trigonometric Ratios!1 a B " A = " A = " A = Sin Cos Tan!1!1 c b c a b a Side opposite of angle A C b Hypotenuse c Side adjacent of angle A A

21 Example 3m N M 5 m P! Find angle P using the proper inverse ratio The length of the hypotenuse is given The opposite measurement of angle P is given What Trigonometric ratio will we use? Sine

22 Solution M SinP = p n 3m N 5 m P SinP = 3 5 SinP = 0.6!P = 0.6 Sin!P = Sin "1 (0.6)!P = " 37 Remember 1/a equals a -1 Therefore, Angle P has an angle of approximately of 37 degrees.

23 Example! The Empire State Building height is m. What is the angle of elevation if a person is 267 m away from the base of the building?

24 Solution Base Top of Building P 267 Person TanP = Opp Adj TanP = TanP =1.428!P = Tan!P = Tan "1 (1.428)!P = " 55 Therefore, angle of elevation is approximately 55 degrees

25 Make sure you have a calculator! Given Ratio of sides Angle, side Looking for Angle measure Missing side Use SIN -1 COS -1 TAN -1 SIN, COS, TAN Set your calculator to Degree!.. MODE (next to 2 nd button) Degree (third line down! highlight it) 2 nd Quit