Chapter 8 Test Wednesday 3/28

Transcription

1 Chapter 8 Test Wednesday 3/28 Warmup Pg. 487 #1-4 in the Geo book 5 minutes to finish 1

2 x = x = x = x = -5 What are we learning today? Pythagoras The Rule of Pythagoras Using Pythagoras Theorem Pythagorean Triples Proving Pythagoras Theorem Converse of the Pythag Theorem 2

3 Pythagoras of Samos - Born B.C. and died B.C. -A Greek mathematician and philosopher known as the father of numbers - believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles - his followers were known as Pythagoreans - He was not the first to use the Pythagorean theorem but he was the first to construct a formal proof of the theorem even though there is no written evidence of this The Rule of Pythagoras The Rule of Pythagoras (pythag. Theorem) The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs of a right triangle. 3

4 Using Pythagoras Theorem You must remember that you can only use the theorem to find a missing side length if you know 2 sides of a right triangle x x x x 13 Using Pythagoras Theorem Pythagoras Theorem can also be used to find the distance between two points in a coordinate plane x x x x

5 Pythagorean Triples If the sides of a right triangle are positive integers the sides are said to be a Pythagorean triple. Pythagorean Triples 5

6 Proving Pythagoras Theorem There are many different proofs for Pythagoras Theorem. Many of them involve area formulas that you already know. Proving Pythagoras Theorem There are many different proofs for Pythagoras Theorem. Many of them involve area formulas that you already know. 6

7 Proving Pythagoras Theorem Converse of the Pythag Theorem If you know the three side lengths of a triangle you can use the Pythagorean Theorem to classify the triangle as acute, right, or obtuse Remember the two smaller sides must have a sum greater than the third side for the triangle to exist! 7

8 Converse of the Pythag Theorem Classify the triangles with the given side lengths as acute, right, or obtuse, if possible this can be a triangle 10,6, this is a right triangle 12,15,28 14, 15, this can't be a triangle this can be a triangle this is an acute triangle Homework pg. 203 #1-15,25-31 in the chapter 8 packet 8

9 Chapter 8 Test Wednesday 3/28 Warmup Pg. 535 #5-8 in the Geo book 5 minutes to finish Please have your homework out 9

10 = x 2 x = 452 x = = x 2 x = 289 x = = x 2 x = 288 x = x 2 = 18 2 x = 243 x =

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12 What are we learning today? Trigonometric Ratios Finding ratios with sine, cosine, and tangent Using trig ratios to find side lengths Inverse trig ratios 12

13 Trigonometric Ratios Trigonometric ratios are ratios using the side lengths of a right triangle. There are 3 basic trig ratios -Sine - Cosine - Tangent Trigonometric Ratios Opposite Adjacent The sine ratio sin opposite hypotenuse The cosine ratio cos adjacent hypotenuse The tangent ratio opposite tan adjacent 13

14 Math Mnemonics: Trigonometry Writing ratios with sine, cosine, and tangent To properly write the ratios using sine, cosine and tangent you must know which sides to pick. The sides are always in relation to an acute angle in the right triangle sin R sin A 25 14

15 Using trig ratios to find side lengths You can use the trig ratios to set up proportions to find the lengths of the sides of a triangle. 1. Pick an acute angle to focus on 2. Label the sides opp. adj. and hyp. 3. Pick which trig ratio corresponds to the side you need to find and the sides you already have. 4. Set up a proportion using your side lengths and trig ratio and then solve the proportion Using trig ratios to find side lengths Solve for the missing variables o w o s tan 20 cos o o w 9 tan 20 s 34cos 23 s 9(0.3640) 3.3 s 34(0.9205) 31.3 o 9 o t sin 70 sin 23 v o t 34sin 23 v o sin t 34(0.3907) 13.3 v

16 o HI cos HI 23cos 53 HI 23(0.6018) 13.8 o IJ sin IJ 23sin 53 IJ 23(0.7986) 18.4 o o Find the missing side lengths o OM tan OM 13tan 27 OM 13(0.5095) 6.6 o 13 cos 27 MP MP o cos MP 14.6 o Classwork pg. 510 #14,17,19 in the geo book 16

17 sin 35 o = x/ = x/20 x = 11.5 sin 36 o = 10/x.5878 = 10/x x = 17.0 tan 25 o = 10/x.4663 = 10/x x = 21.4 Homework pg. 211 #1, 4-10 in the ch 8 packet pg. 510 #15,16,18 in the geo book 17

18 Chapter 8 Test Wednesday 3/28 Warmup Pg. 213 #1,2,4,6 in the Workbook 5 minutes to finish Please have your homework out 18

19 C Tan 75 o = x/22 x = 82.1 i cos 22.2 o = x/14 x = G cos 34 o = 17/x x = 20.5 F 19

20 cos 41 o = 11/x x = 14.6 tan 64 o = x/7 x = 14.4 sin 28 o = 50/x x =

21 What are we learning today? Inverse trig ratios Angles of Elevation Angles of Depression Trigonometric Ratios The Sine ratio Sine = Opposite / Hypotenuse Sin = Opp / Hyp The Cosine ratio Cosine = Adjacent / Hypotenuse Cos = Adj / Hyp The Tangent ratio Tangent = Opposite / Adjacent Tan = Opp / Adj 21

22 Using trig ratios to find side lengths You can use the trig ratios to set up proportions to find the lengths of the sides of a triangle. 1. Pick an acute angle to focus on 2. Label the sides opp. adj. and hyp. 3. Pick which trig ratio corresponds to the side you need to find and the sides you already have. 4. Set up a proportion using your side lengths and trig ratio and then solve the proportion x z y o y tan o y 40 tan o z tan o z 40 tan x ft x z y 22

23 Inverse Trig Ratios If Sin A = opp/hyp, then Sin -1 (opp/hyp) = m A If Cos A = adj/hyp, then Cos -1 (adj/hyp) = m A If Tan A = opp/adj, then Tan -1 (opp/adj) = m A tan B = 2/3 tan -1 (2/3) = B B = 33.7 A = A = 56.3 Solving Right Triangles When you are asked to solve a triangle you must find the length of every side and the measure of every angle. 23

24 Solve the triangle UM UM cos E 28 cos E E cos (0.6786) E 47.3 o U o Solve the triangle o V o 14 sin 51 DV 14 DV 18.0 o sin 51 o 14 tan 51 DM 14 DM 11.3 o tan 51 24

25 Classwork pg. 511 #22-24 sin x = 5/14 x = sin -1 (.3571) x = 21 o tan x = 8/5 x = tan -1 (1.6) x = 58 o cos x = 9/13 x = cos -1 (.6923) x = 46 o 25

26 Angles of Elevation and Depression An angle of elevation is an angle measured above the horizontal and an angle of depression is an angle measured below the horizontal Suppose you stand 53 ft from a wind farm turbine. Your angle of elevation to the hub of the turbine is 56.5 o. Your eye level is 5.5 ft above the ground. Approximately how tall is the turbine from the ground to its hub? 26

27 To approach runway 17 of the Ponca City Municipal Airport in Oklahoma, the pilot must being a 3 o descent starting from a height of 2714 ft above sea level. The airport is 1007 ft above sea level. To the nearest tenth of a mile, how far from the runway is the airplane at the start of this approach? One of the wind turbines that Yoshio owns is shown in the figure below. He needs to know the height of the turbine. Yoshio is 5 feet 9 inches tall. If he stands 12 feet from the turbine, his line of sight forms a 90 o angle with the top and bottom of the turbine. What is the height of the turbine? XW XW 144 XW 26.2 ft XY XY 31.7 ft 27

28 Angles of Elevation and Depression Classwork pg. 519 #20-22 sin 27 o = x/580 x = 580sin 27 o x = yd tan 18 o = x/2 x = 2tan 18 o x =.6 km tan 18 o = 250/x x = 250/tan 18 o x = 769 ft 28

29 Homework pg. 211 #11-14 pg. 215 #2-20 even in the chapter 8 packet Chapter 8 Test Wednesday 3/28 29

30 Warmup Pg. 536 #16-18 in the Geo book 5 minutes to finish Please have your homework out tan 36 o = 12/x x = 12/tan 36 x = 16.5 sin x = 12/22 x = sin -1 (12/22) x = 33.1 o sin 58 o = x/45 x = 45 sin 58 o x = 38.2 ft 30

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33 What are we learning today? Special Right Triangles Triangles Triangles Special Right Triangle Shortcuts Law of Sines Law of Cosines Special Right Triangles Special right triangles are right triangles that have the following angle measures 45 o 45 o 90 o And 30 o 60 o 90 o You can find the side lengths of these special right triangles using proportions. Each one of the special right triangles has their own ratio of sides. 33

34 Triangles The ratio of the sides of a right triangle is 1 : 1 : Triangles The ratio of the sides of a right triangle is 1 : 1 : x x y y

35 Triangles The ratio of the sides of a right triangle is 1 : 3 : Triangles The ratio of the sides of a right triangle is 1 : 3 : x x y y

36 Shortcuts with Special right triangles x 4*2 x 8 y x y x 10 2 y

37 6 x 3 2 y 3 3 x 12*2 x 24 y 12 3 x y 12 2 x 26cm x x x 30.0cm 3 37

38 Trigonometric Ratios The Sine ratio Sine = Opposite / Hypotenuse Sin = Opp / Hyp The Cosine ratio Cosine = Adjacent / Hypotenuse Cos = Adj / Hyp The Tangent ratio Tangent = Opposite / Adjacent Tan = Opp / Adj Law of Cosines In any triangle you can use the law of cosines to find a side length or an angle measure if you know: 1) 2 side lengths and an included angle 2) 3 side lengths a 2 = b 2 + c 2 2(b)(c)(cos A) b 2 = a 2 + c 2 2(a)(c)(cos B) c 2 = a 2 + b 2 2(a)(b)(cos C) 38

39 Law of Cosines x x x *27*23*cos x o Classwork pg. 523 #9,10 only find the x value 39

40 x x x *80* 78* cos x o x x x *5*9* cos x o Law of Sines In any triangle you can use the law of sines to find a side length or an angle measure if you know: 1) 2 side lengths and a non included angle 2) 1 side length and 2 angle measures 40

41 Law of Sines x 12 o sin x sin 38 o o sin110 sin x 12 sin x sin x x sin x x x sin o Homework pg #1-6,18,19 pg. 212 #18 in the chapter 8 packet and Trig Worksheet on the class website 41

42 Chapter 8 Test Tomorrow Warmup Pg. 536 #16-18 in the Geo book 5 minutes to finish Please have your homework out 42

43 tan 36 o = 12/x x = 12/tan 36 x = 16.5 sin x = 12/22 x = sin -1 (12/22) x = 33.1 o sin 58 o = x/45 x = 45 sin 58 o x = 38.2 ft 43

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46 Please get into groups for the review game 46