Geometry. Trigonometry of Right Triangles. Slide 1 / 240. Slide 2 / 240. Slide 3 / 240

Transcription

1 New Jersey enter for Teaching and Learning Slide 1 / 240 Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. lick to go to website: Slide 2 / 240 Geometry Trigonometry of Right Triangles Slide 3 / 240 Table of ontents Pythagorean Theorem Similarity in Right Triangles Special Right Triangles Trigonometric Ratios Solving Right Triangles ngles of Elevation and Depression Law of Sines and Law of osines rea of an Oblique Triangle lick on a Topic to go to that section

2 Slide 4 / 240 Pythagorean Theorem Return to the Table of ontents Slide 5 / 240 efore learning about similar right triangles and trigonometry, we need to review the Pythagorean Theorem and the Pythagorean Theorem onverse. Slide 6 / 240 Recall that a right triangle is a triangle with a right angle. hypotenuse leg leg The sides form that right angle are the legs. The side opposite the right angle is the hypotenuse. The hypotenuse is also the longest side.

3 Slide 7 / 240 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. leg2 + leg2 = hypotenuse2 or a2 + b2 = c2 a c b Slide 8 / 240 Eample: 9 12 Find the length of the missing side of the right triangle. Is the missing side a leg or the hypotenuse of the right triangle? Slide 8 () / 240 Eample: Find the length of the missing side of the right triangle. hypotenuse 9 = [This object is of a pullthe tab] right triangle? Is the missing side a leg or the hypotenuse

4 Slide 9 / 240 Solve for : = = = = -15 is a etraneous solution, a distance can not equal a negative number. = 15 Slide 10 / 240 Eample: Is the missing side a leg or the hypotenuse of the right triangle? Find the length of the missing side of the right triangle. Slide 10 () / 240 Eample: Find the length of the missing side of the right triangle. leg 28 Is the missing side a leg or the hypotenuse of the right triangle? 20

5 Slide 11 / The missing side is the of the right triangle. hypotenuse 6 9 leg Slide 11 () / The missing side is the of the right triangle. hypotenuse leg 6 9 Slide 12 / Find the length of the missing side. 6 9

6 Slide 12 () / Find the length of the missing side. 6 9 Slide 13 / 240 leg hypotenuse The missing side is the of the right triangle. hypotenuse leg The missing side is the of the right triangle. Slide 13 () / 240

7 Slide 14 / Find the length of the missing side Slide 14 () / Find the length of the missing side Slide 15 / 240 Real World pplication The safe distance of the base of the ladder from a wall it leans against should be one-fourth of the length of the ladder. Thus, the bottom of a 28-foot ladder should be 7 feet from the wall. How far up the wall will a ladder reach?? 28 feet 7 feet

8 Slide 16 / ? 28 feet 7 feet The ladder will reach 2 Solve using a + b = c feet up the wall safely. Slide 16 () / Solve using a + b = c? 28 feet 7 feet The ladder will reach feet up the wall safely. Slide 17 / 240 Real World pplication 50 The dimensions of a high school basketball court are 84' long and 50' wide. What is the length from one corner of the court to the opposite corner? 84

9 Slide 17 () / 240 Real World pplication = = = The court is feet The dimensions of a high school basketball court are 84' long and 50' wide. What is the length from one corner of the court to the opposite corner? Slide 18 / N court is 50 feet wide and the length from one corner of the court to the opposite corner is feet. How long is the court? (Round the answer to the nearest whole number) feet feet 118 feet D 94 feet 5 N court is 50 feet wide and the length from one corner of the court to the opposite corner is feet. How long is the court? feet feet 118 feet (Round the answer to the nearest whole number) D D 94 feet Slide 18 () / 240

10 Slide 19 / 240 Pythagorean Theorem pplications The Pythagorean Theorem can also be used in figures that contain right angles. Slide 20 / 240 Eample Find the perimeter of the square. 18 cm Psq = 4s note: efore finding the perimeter of the square, we need to first find the length of each side. Slide 21 / 240 Remember, in a square all sides are congruent. Start here: = cm

11 Slide 21 () / 240 Remember, in a square all sides are congruent. Start here: = cm Slide 22 / 240 Eample Find the area of the triangle. The base of the triangle is given, but we need to find the height of the triangle. = 1 2 bh 13 feet 13 feet 10 feet Slide 23 / feet h 5 feet 13 feet 5 feet y definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base.

12 Slide 23 () / 240 y definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. 13 feet h 5 feet 13 feet 5 feet Slide 24 / 240 Try this... Find the perimeter of the rectangle. P 8 in = 2l + 2w rect 10 in Slide 24 () / 240 Try this... Find the perimeter of the rectangle. P 8 in 10 in rect = 2l + 2w = 102 =6 Prect = 2(6) + 2(8) P = 28 inches

13 Slide 25 / Find the area of the rectangle. 84 square feet 46 square inches 8 feet 17 fe e t 120 square feet D 46 square feet Slide 25 () / Find the area of the rectangle. 120 square feet 84 square feet 46 square inches 8 feet 17 fe e t D 46 square feet Slide 26 / Find the perimeter of the square. (Round to the nearest tenth) D 36 cm cm 25.6 cm cm 12.8 cm

14 Slide 26 () / Find the perimeter of the square. (Round to the nearest tenth) 12.8 cm D 36 cm cm 25.6 cm cm Slide 27 / inches 7 inches 8 Find the area of the triangle. 10 inches Slide 27 () / Find the area of the triangle. 7 inches 7 inches h=4.9in =24.5 square inches 10 inches

15 Slide 28 / Find the area of the triangle. 7 inches 7 inches 4 inches Slide 28 () / Find the area of the triangle. 7 inches 7 inches h=6.7in =13.4 square inches 4 inches Slide 29 / 240 onverse of the Pythagorean Theorem If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If c = a + b, then is a right triangle a c b

16 Slide 30 / 240 Eample Tell whether the triangle is a right triangle. 24 Remember c is the longest side E 7 25 D F Slide 30 () / 240 Eample Tell whether the triangle is a right triangle. 24 E D is a right triangle. If c = a + b, then If then is a right triangle. 7 Remember c is the therefore is a right triangle. longest side 25 F Slide 31 / 240 Theorem If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. If c > a + b, then is obtuse. 2 2 c a 2 b

17 Slide 32 / 240 Theorem If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. a c If c2 < a2 + b2, then is acute. b Slide 33 / 240 Eample lassify the triangle as acute, right, or obtuse Slide 33 () / 240 Eample lassify the triangle as acute, right, or obtuse. c = Since 17 the triangle is acute. 13

18 Slide 34 / lassify the triangle as acute, right, obtuse, or not a triangle. acute obtuse 12 right 15 D not a triangle lassify the triangle as acute, right, obtuse, or not a triangle. Slide 34 () / 240 right obtuse D not a triangle acute Slide 35 / 240 acute right obtuse D not a triangle lassify the triangle as acute, right, obtuse, or not a triangle.

19 11 lassify the triangle as acute, right, obtuse, or not a triangle. right obtuse acute Slide 35 () / D not a triangle 6 acute 25 right obtuse 20 D not a triangle lassify the triangle as acute, right, obtuse, or not a triangle. obtuse D not a triangle acute right Slide 36 / lassify the triangle as acute, right, obtuse, or not a triangle Slide 36 () / 240

20 13 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. Slide 37 / 240 acute right obtuse 13 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. Slide 37 () / 240 right obtuse acute Slide 38 / 240 acute triangle right triangle obtuse triangle 14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle.

21 14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. Slide 38 () / 240 right triangle acute triangle obtuse triangle Slide 39 / 240 Review If c = a + b, then triangle is right If c2 > a2 + b2, then triangle is obtuse. If c2 < a2 + b2, then triangle is acute. Slide 40 / 240 Similarity in Right Triangles Return to the Table of ontents

22 Slide 41 / 240 There are many proofs to the Pythagorean Theorem. How many do you know? Triangle similarity can be used to prove the Pythagorean Theorem. How? Slide 42 / 240 Theorem The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. D D is the altitude of ~ D~ D Slide 43 / 240 Teacher Notes To prove this, click for Lab 1 - Similar Right Triangles Therefore, the altitude of a right triangle divides the triangle into two smaller triangles that are similar to the click original triangle and similar to each other. click

23 Slide 43 () / 240 To prove this, click for Lab 1 - Similar Right Triangles See Similar Right Triangles Lab tocomplete this activity. Teacher Notes 1. ut an inde card along one of its diagonals, forming two congruent right triangles. 2. For one right triangle, draw an altitude from the right angle to the hypotenuse. ut along the altitude to form two right triangles. 3. ompare the three triangles. What special property do they share? [This object is a pull tab] Therefore, the altitude of a right triangle divides the triangle into two smaller triangles that are similar to the click original triangle and similar to each other. click The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. Given: Prove: D~ D Statements Reasons Given click is a right triangle is a right triangle is the altitude of ~ Slide 44 / 240 Let's prove the Theorem. is a right angle Given click Def clickof ltitude D Def of Perp Lines. 2 lines that form a rtclick angle is a right angle ll rt angles are click Refleive Prop of click ~ D ~ click is a right angle Def of Perp Lines click ll rt angles are click Refleive Prop of click ~ D ~ D~ D D Transitive Prop of ~ click Slide 45 / 240 Let's sketch the 3 triangle's separately, with the same orientation. ~ click D Match up the angles. D Helpful tip: If you set, then you can assign all the angles a value and easily find the matches D D

24 e c ssign lengths to all the segments. Let the lengths of the segments on the hypotenuse be d and e. Label the sides of a triangle with the lower case letter of the opposite angle. a c d b Slide 46 / 240 a b a b d D e ~ D D~ D ecause the triangles are similar the corresponding sides are proportional. ~ To prove the Pythagorean Theorem, use the proportions. Given: ~ D ~ D click Using the multiplication property of equality, multiply the equation by bc. click (1) ~ a D simplify click ltitude of a rt triangle click theorem. Definition of similar triangles. click d Using the multiplication property of equality, b click multiply the equation by ac (2) To prove the Pythagorean Theorem, use the proportions (continued). Given: Slide 47 / 240 ltitude of a rt triangle click theorem. Definition of similar triangles. Prove: c Reasons Statements is a right triangle. is an altitude. e D is a right triangle. is an altitude. Prove: Statements simplify click Reasons Using the addition property of equality, add equation (1) and equation (2) together. click Distributive Property click click Given Substitution click e c d b a Simplify click Slide 48 / 240

25 Slide 49 / 240 Eample Find the length of the altitude KI? H I K H J 5 Slide 50 / 240 It maybe helpful to sketch the 3 triangle's separately, with the same orientation. H H I K K 5 J 5 12 J K I 13 I J 12 5 K ecause the triangles are similar the corresponding sides are proportional. 13 = Slide 51 / 240 Try this... Find the length of RS. 3 P Q R P 4 3 R R S Q S 4 Q P 5 S S

26 Slide 51 () / 240 Try this... Find the length of RS. 3 P Q R 4 5 R S 5 P 4 3 S Q S 4 Q P 5 R S Slide 52 / Which ratio is the ratio of corresponding sides? I H K J D 15 Which ratio is the ratio of corresponding sides? I H K J D Slide 52 () / 240

27 Slide 53 / Find KJ. I 7 H 24 J K 25 Slide 53 () / Find KJ. I 7 H 24 Set KJ = J K 25 The net two theorems are Geometric Mean Theorems. What is a mean? n average. Usually when we ask to find the mean, we are asking for the arithmetic mean. What is an arithmetic mean? The sum of n values divided by the number of values (n). What is a geometric mean? The nth root of a product of n values. It is defined for only positive numbers (no negative numbers, no zero) For more information click on this link: rithmetic Mean vs Geometric Mean Slide 54 / 240

28 Slide 55 / 240 The geometric mean of two positive numbers a and b is the positive number that satisfies a = b 2 = ab = Visually, the geometric mean answers this question: given a rectangle with sides a and b, find the side of the square whose area equals that of the rectangle. Slide 56 / 240 Eample Find the geometric mean of 8 and = 8(14) 2 = 112 (only the positive value) Slide 57 / Find the geometric mean of 7 and 56. Write the answer is simplest radical form. D

29 Slide 57 () / Find the geometric mean of 7 and 56. Write the answer is simplest radical form. D Slide 58 / Find the geometric mean of 3 and 48. Students type their answers here 18 Find the geometric mean of 3 and 48. Students type their answers here Slide 58 () / 240

30 Slide 59 / 240 orollary The altitude drawn to the hypotenuse of a right triangle divides the the hypotenuse into two segments. The altitude is the geometric mean of the two segments formed. D is the altitude of Since, D~ D D D2 = D(D) Slide 60 / 240 Eample Find z. 8 6 z Slide 60 () / 240 Eample Find z. 8 6 z

31 Slide 61 / 240 Eample Find z z Slide 61 () / 240 Eample Find z z Slide 62 / 240 Try this... Find y y 2) y )

32 Slide 62 () / 240 Try this... Find y. 12 y 9 1) y 2) ) y = 12 2) y = 6.75 Slide 63 / Find D 50 Slide 63 () / Find D

33 Slide 64 / Find D Slide 64 () / Find D orollary If the altitude drawn to the hypotenuse of a right triangle, divides the hypotenuse into two segments. The length of each leg of the original triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. D D is the altitude of Since, ~ ~ D D D = D~ ~ D = D Slide 65 / 240

34 Eample 4 R S Slide 66 / T 9 T Find. U Eample 4 R S Slide 66 () / 240 Find. U Slide 67 / 240 Eample D E 4 F 6 G Find.

35 Slide 67 () / 240 Eample Find. D 4 E F 6 G Slide 68 / Is PR a geometric mean between QR and SR? False P True Q R S 21 Is PR a geometric mean between QR and SR? P False True S Q False R Slide 68 () / 240

36 Slide 69 / Is the geometric mean correct? P False True Q R S Slide 69 () / Is the geometric mean correct? P True False Q True R S Slide 70 / Which proportion is correct? K L M D J

37 Slide 70 () / Which proportion is correct? K L J M D Slide 71 / Find y y 16 D 12 Slide 71 () / Find y y D 9 16 D 12

38 Slide 72 / Find y y 24 9 D None of the above 3 Slide 72 () / Find y y 9 D None of the above Slide 73 / Find. 5 8

39 Slide 73 () / Find. 5 8 Slide 74 / 240 Special Right Triangles Return to the Table of ontents In this section you will learn about the properties of the two special right triangles o 90o 60o 45o 90o 30o Slide 75 / 240

40 Slide 76 / Triangle Theorem triangle is an isosceles right triangle, where the hypotenuse is 2 times the length of the leg. hypotenuse = leg( 2) 45o an you prove this? 2 45o Slide 76 () / Triangle Theorem triangle is an isosceles right triangle, where the hypotenuse is 2 times the length of the leg. an you prove this? hypotenuse = leg( 2) 45o 2 45o Slide 77 / 240 P 6 Q 45 o y 45o R Eample Find the length of the missing sides. Write the answer in simplest radical form.

41 Slide 77 () / 240 Eample Find the length of the missing sides. Write the answer in simplest radical form. 6 y the orollary to the P ase ngles 45 Thm, PQ=QR. o y=6 Q hypotenuse = 2(leg) = 2(6) = 6 2 y 45o R y S Slide 78 / 240 T 18 Eample Find the length of the missing sides of the right triangle. V Eample Find the length of the missing sides of the right triangle. Since, STU is an isosceles rt triangle S hypotenuse = leg y Slide 78 () / 240 T ST=TV =y There are 2 ways to solve. 18 V

42 Slide 79 / 240 Try this... Find the length of the missing sides. y 8 Slide 79 () / 240 Try this... Find the length of the missing sides. y 8 Slide 80 / Find the value of. 5 (5 2)/2 y 5 5 2

43 Slide 80 () / Find the value of. 5 (5 2)/2 5 2 y 5 Slide 81 / Find the value of y. 5 2 y 5 (5 2)/2 Slide 81 () / Find the value of y. (5 2)/ y 5

44 Slide 82 / What is the length of the hypotenuse of an isosceles right triangle, if the length of the legs is 8 2 inches. 29 What is the length of the hypotenuse of an isosceles right triangle, if the length of the legs is 8 2 inches. ) hypotenuse = leg( Slide 82 () / 240 Slide 83 / What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm.

45 Slide 83 () / What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm.= leg( ) hypotenuse Slide 84 / Triangle Theorem In a right triangle, the hypotenuse is twice the length of the shorter leg and the longer leg is 3 times the length of the shorter leg. 60o 2 30o 3 hypotenuse = 2(shorter leg) longer leg = 3(shorter leg) Slide 85 / 240 This can be proved using an equilateral triangle. For right triangle D, D is a perpendicular bisector. let a =, c = 2 and b= D c=2 60 a= b D 60o 60 30o

46 Slide 86 / 240 G 30o Eample Find the length of the missing sides of the right triangle. y 60o H 5 30o 60o y = F Slide 87 () / 240 G 30o GF is the longest side (hypotenuse) GH is the 2nd longest longerside leg = 3(shorter leg) HF < GH < GF y = 3(5) y H Recall triangle inequality, the shortest side is opposite the smallest angle and the longest hypotenuse = 2(shorter leg) side is opposite the largest angle. = 2(5) = 10 HF is the shortest side Slide 87 / 240 G Recall triangle inequality, the shortest side is opposite the smallest angle and the longest side is opposite the largest angle. HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF F y H 60o 5 F

47 M 60o y 9 Slide 88 / 240 Eample Find the length of the missing sides of the right triangle. 30o T Eample Find the length of the missing sides of the right triangle. M 60o M is the shorter leg and MT is the longer leg 9 Slide 88 () / 240 y longer leg = 3(shorter leg) 30o 9 = 3() 3 3 = T Slide 89 / 240 Eample Find the area of the triangle. 14 ft

48 Slide 90 / 240 The altitude (or height) divides the triangle into two 30o-60o-90o triangles. 14 ft h?? The length of the shorter leg is 7 ft. The length of the longer leg is 7 3 ft. = b(h) = 14(7 3) square ft Slide 91 / 240 Try this... Find the length of the missing sides of the right triangle o y 60 o Slide 91 () / 240 Try this... Find the length of the missing sides of the right triangle o y 60 o

49 Slide 92 / 240 Try this... Find the area of the triangle. 9 ft 30o Slide 92 () / 240 Try this... Find the area of the triangle. 9 ft 30o short leg = 4.5 ft long leg = 4.5 ft Slide 93 / Find the value of. 7 3 (7 2)/2 D 14 60o 7 30o 7

50 Slide 93 () / Find the value of. 7 3 (7 2)/2 7 60o 7 30o D 14 Slide 94 / Find the value of (7 2)/2 7 D 14 Slide 94 () / Find the value of. 7 3 (7 2)/2 7 D D

51 Slide 95 / Find the value of o o (7 2)/2 D 14 Slide 95 () / Find the value of. 7 (7 2)/2 D o 60o Slide 96 / The hypotenuse of a 30o -60o -90o triangle is 13 cm. What is the length of the shorter leg?

52 Slide 96 () / The hypotenuse of a 30o -60o -90o triangle is 13 cm. What is the length of the shorter leg? shorter leg = 13/2 shorter leg = 6.5cm Slide 97 / The length the longer leg of a 30o -60o -90o triangle is 7 cm. What is the length of the hypotenuse? 35 The length the longer leg of a 30o -60o -90o triangle is 7 cm. What is the length of the hypotenuse? shorter leg = hypotenuse = = Slide 97 () / 240

53 Slide 98 / 240 Real World Eample The wheelchair ramp at your school has a height of 2.5 feet and rises at angle of 30o. What is the length of the ramp? Slide 99 / 240? o The triangle formed by the ramp is a 30o-60o-90o right triangle. The length of the ramp is the hypotenuse. hypotenuse = 2(shorter leg) hypotenuse = 2(2.5) hypotenuse = 5 The ramp is 5 feet long. Slide 100 / o? 3 feet 36 skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45. What is the height of the ramp? Round to the nearest hundredth.

54 Slide 100 () / skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45. What is the height of the ramp? Round to the nearest hundredth. 45o 3 feet? Slide 101 / o 3 feet 37 What is the length of the base of the ramp? Round to the nearest hundredth.? 37 What is the length of the base of the ramp? Round to the nearest hundredth. 45o 3 feet? Slide 101 () / 240

55 38 The yield sign is shaped like an equilateral triangle. Find the length of the altitude. Slide 102 / inches 38 The yield sign is shaped like an equilateral triangle. Find the length of the altitude. Slide 102 () / inches Slide 103 / The yield sign is shaped like an equilateral triangle. Find the area of the sign. 20 inches

56 39 The yield sign is shaped like an equilateral triangle. Find the area of the sign. Slide 103 () / inches Slide 104 / 240 Trigonometric Ratios Return to the Table of ontents Slide 105 / 240 Right triangle trigonometry is the study of the relationships between the sides and angles of right triangles. c b a

57 Slide 106 / 240 Ever since the construction of the ell Tower in the 1100's, it has slowly tilted south and is at risk of falling over. If the angle of slant ever fall's below 83 degrees, it is feared the tower will collapse. Leaning Tower of Pisa, ell Tower in Pisa, Italy Slide 107 / 240 Engineers can measure the angle of slant using any of the right triangles constructed below. Engineers very carefully measure the perpendicular distance from a tower window (points, D or F) to the ground (points G, E or ). Then they measure the distance from the tower to points, E or G. D F ~ DE~ FG WHY? GE angle of slant Slide 107 () / 240 Engineers can measure the angle of slant using any of the right triangles constructed below. Engineers very carefully measure the perpendicular distance from a tower window (points, D or F) to the ground (points G, E or ). Then they measure the distance ~ from the tower to points, E or G. ~ DE~ FG D F WHY? GE angle of slant

58 Slide 108 / 240 Triangle Height ase Ratio Height / ase =50m =5m 50/5=10 DE DE=30m E=3m 30/3=10 FG FG=20m G=2m 20/2=10 Let's calculate the ratio's of the height to the base for each right triangle. Notice that all of the ratios are the same. WHY? The ratio of height/base is also called the slope ratio (rise/run) or tangent ratio. Slide 109 / 240 When the triangle is dilated (pull scale), how does the angle change? What happens to the slope ratio? What happens to the ratio when the angle increases? What happens to the ratio when the angle decreases? lick for interactive website to investigate. Slide 110 / 240 To learn right triangle trigonometry, first you need to be able to identify the sides of a right triangle. In a right triangle, there are 2 acute angles. In the triangle to the left, and are the acute angles. Label the sides of a triangle with the lower case letter of the opposite angle. c b a

59 Slide 111 / 240 Let's look at, when is the reference angle, the side opposite is a. the side adjacent (or net to) is b. b and the hypotenuse is c. adj c b adj a hyp hyp a opp c opp When is the reference angle, the side opposite is b. the side adjacent (or net to) is a. and the hypotenuse is c. Slide 112 / What is the side opposite to J? L J LK KJ K Slide 112 () / What is the side opposite to J? JL KJ L J LK JL K

60 Slide 113 / What is the hypotenuse of the triangle? L J LK KJ K JL Slide 113 () / What is the hypotenuse of the triangle? JL KJ LK L J K Slide 114 / What is the side adjacent to J? JL J L KJ K LK

61 Slide 114 () / What is the side adjacent to J? JL L J LK KJ K Slide 115 / What is the side opposite K? L J LK K KJ Slide 115 () / What is the side opposite K? L J KJ JL LK JL K

62 Slide 116 / What is the side adjacent to K? L J LK JL K KJ Slide 116 () / What is the side adjacent to K? L J JL KJ LK K Slide 117 / 240 Trigonometric Ratios trigonometric ratio is the ratio of the two sides of a right triangle. There are 3 ratios for each acute angle of a right triangle. The ratios are called sine, cosine, and tangent (abbreviated sin, cos, and tan). c b a

63 Slide 118 / 240 The 3 Trigonometric Ratios sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse opposite side adjacent side tanθ = This spells... SOHHTO θ or c b which is a pneumonic to help you remember the sides of a right triangle (you'll need to remember the spelling). a Slide 119 / 240 lick for a SOHHTO song on youtube.com "Gettin' Triggy Wit It". Slide 120 / 240 D Eample Find the sin F, cos F, and tan F E F 8 Since F is your reference angle, label the sides of the triangle opposite, adjacent and hypotenuse. Use the pneumonic to find the trig ratios. lways reduce fractions to lowest terms. D opp 10 6 E adj 8 sinf hyp F opp = hyp 3 6 = 10 = 5 cosf = adj = 8 = 4 hyp 10 5 tan F = opp = 6 = 3 adj 8 4

64 Slide 121 / 240 D Eample Find the sin D, cos D, and tan D E F 8 Since D is your reference angle, label the sides of the triangle opposite, adjacent and hypotenuse. Use the pneumonic to find the trig ratios. lways reduce fractions to lowest terms. D adj 10 6 E opp 8 sind hyp F opp 8 4 = hyp = 10 = 5 cosd = adj = 6 = 3 hyp tand = opp = 8 = 3 6 adj Slide 122 / What is the sin R? 20/29 21/20 21/29 D 20/21 Slide 122 () / What is the sin R? 21/20 21/29 20/29 D 20/21

65 Slide 123 / What is the cosr? 20/29 21/20 21/29 D 20/21 Slide 123 () / What is the cosr? 21/20 21/29 D 20/21 20/29 Slide 124 / What is the tanr? 21/20 20/29 D 21/29 20/21

66 Slide 124 () / What is the tanr? 20/21 21/20 20/29 D 21/29 Slide 125 / What is the sinq? 20/29 21/20 21/29 D 29/20 Slide 125 () / What is the sinq? 21/20 21/29 20/29 D 29/20

67 Slide 126 / What is the cosq? 20/29 21/20 21/29 D 29/21 Slide 126 () / /29 21/20 49 What is the cosq? 21/29 D 29/21 Slide 127 / What is the tanq? 21/20 21/29 D 20/21 20/29

68 Slide 127 () / What is the tanq? 20/29 21/29 D 21/20 D 20/21 Slide 128 / 240 The angle of slant of the Tower of Pisa is 84.3 To find the trigonometric ratio of an angle, use a calculator or a trig table. heck that your calculator is set for degrees (not radians) and round your answer to the ten thousandth place (4 decimal places). D F Find the following: click sin 84.3 =.9951 cos 84.3 =.0993 click tan 84.3 = click angle of slant Slide 129 / D Evaluate sin 60. Round to the nearest ten thousandth.

69 Slide 129 () / Evaluate sin 60. Round to the nearest ten thousandth D Slide 130 / Evaluate cos 60. Round to the nearest ten thousandth D Evaluate cos 60. Round to the nearest ten thousandth D Slide 130 () / 240

70 Slide 131 / Evaluate tan 60. Round to the nearest ten thousandth D Slide 131 () / Evaluate tan 60. Round to the nearest ten thousandth D Slide 132 / 240 Trig tables were used by early mathematicians and astronomers to calculate distances that they were unable to measure directly. Today, calculators are usually used.

71 How do you find an unknown side measure in a right triangle when you are given an acute angle and one side? Slide 133 / 240 You need to identify the correct trig function that will find the missing side. Use SOHHTO to help. is your angle of reference. Label the given and unknown sides of your triangle opp, adj, or hyp. Identify the trig funtion that uses, the unknown side and the given side. Using, I am looking for o and I have a, so the ratio is o/a which is tangent. now you can solve for, the missing side. opp adj Slide 134 / 240 Eample Find the trig equation that will find o Slide 134 () / 240 Eample Find the trig equation that will find o adj 30o 12 opp

72 Slide 135 / 240 Eample Find the trig equation that will find. 30o 12 Slide 135 () / 240 Eample Find the trig equation that will find. 30o 12 adj 30o hyp 12 Slide 136 / 240 Eample Find the trig equation that will find. 30o 12

73 Slide 136 () / 240 Eample Find the trig equation that will find o 30o hyp 12 opp Slide 137 / Using, which is the correct trig equation needed to solve for. sin40 = 12/ o cos40 = /12 40 tan40 = 12/ 12 D sin40 = /12 D E Slide 137 () / Using, which is the correct trig equation needed to solve for. sin40 = 12/ cos40 = /12 tan40 = 12/ 0o 4 12 D sin40 = /12 E D

74 Slide 138 / Using D, which is the correct trig equation needed to solve for. cos50 = /12 12 tan50 = 12/ sin50 = 12/ D sin50 = /12 50o E D Slide 138 () / Using D, which is the correct trig equation needed to solve for. cos50 = /12 tan50 = 12/ sin50 = 12/ D 12 D sin50 = /12 50o E D tan32 = /11 D sin32 = 11/ K 11 cos32 = /11 tan32 = 11/ Slide 139 / 240 J 32o L 56 Using J, which is the correct trig equation needed to solve for.

75 Slide 139 () / 240 tan32 = /11 cos32 = /11 tan32 = 11/ 56 Using J, which is the correct trig equation needed to solve for. K 11 J D sin32 = 11/ 32o L 58 o tan58 = /11 cos58 = /11 tan58 = 11/ J D sin 58 = 11/ tan58 = 11/ D sin 58 = 11/ 58 o cos58 = /11 J K 11 L 57 Using K, which is the correct trig equation needed to solve for. tan58 = /11 Slide 140 / Using K, which is the correct trig equation needed to solve for. Slide 140 () / 240 K 11 L

76 Slide 141 / 240 Finding the Missing Side of a Right Triangle Now, you can solve for, the missing side. Round your answer to the nearest tenth. Using your calculator, find the tan 84.3 Round your answer to 4 decimal places. opp You can rewrite with a denominator of 1 and use the cross product property or multiply both sides of the equation by 5 using the multiplication property of equality (see net slide). adj Slide 142 / 240 Finding the Missing Side of a Right Triangle Now, you can solve for, the missing side. Round your answer to the nearest tenth. opp Multiply both sides of the equation by 5 using the multiplication property of equality. adj Slide 143 / 240 Eample Find. Round your answer to the nearest hundredth. 25o E 12 M G

77 Slide 143 () / 240 Eample Find. Round your answer to the nearest hundredth. E 25o G 12 sin G = EM GM sin25 = 12 (12).4226 = 12 (12) 5.07 M Slide 144 / 240 Eample Find. Round your answer to the nearest hundredth. E G 12 o 65 M Eample Find. Round your answer to the nearest hundredth. 12 E G cos M = EM GM cos 65 = 12 (12).4226 = o (12) 5.07 M Slide 144 () / 240

78 Slide 145 / 240 Eample Find y. Round your answer to the nearest hundredth o E y Slide 145 () / o E y Eample Find y. Round your answer to the nearest hundredth. tan = E E tan 20 = 10 y (y).3640 = 10 y (y).3640y = 10 y Slide 146 / 240 P 12 L 68o M 58 Find the length of LM. Round your answer to the nearest tenth.

79 Slide 146 () / 240 P 58 Find the length of LM. Round your answer to the nearest tenth o 59 Find the length of LP. Round your answer to the nearest tenth. M Slide 147 / 240 P 12 L 68o 59 Find the length of LP. Round your answer to the nearest tenth. M Slide 147 () / 240 P 12 L 68o L M

80 Slide 148 / 240 Eplain and use the relationship between the sine and cosine of complementary angles. Slide 149 / 240 Find the measure of? To find the measure of... The sum of the interior angles of any triangle is equal to 180 degrees. and are complementary angles. omplementary angles are two angles whose sum of their measures is 90 degrees. The acute angles of a right triangle are always complementary. Slide 150 / 240

81 Slide 151 / For right triangle, what is the measure of? 30 degrees 50 degrees 60 degrees D cannot be determined 30o Slide 151 () / For right triangle, what is the measure of? 50 degrees 60 degrees 30 degrees D cannot be determined 61 If the [Thisoobject is a pull tab] 30 Slide 152 / 240, find the complementary angle? 70 degrees 160 degrees D none of the above 20 degrees

82 61 If the Slide 152 () / 240, find the complementary angle? 70 degrees 160 degrees 20 degrees D none of the above Let's compare the sine and cosine of the acute angles of a right triangle. In a right triangle, the acute angles are complementary. m + m = = 90 5 Slide 153 / sin = 4/5 sin 53.1 =.7997 cos = 4/5 cos 36.9 =.7997 sin = cos 53.1 sin 53.1 = cos The sine of an angle is equal to the cosine of its complement. cos = 3/5 cos 53.1 =.6004 sin = 3/5 sin 36.9 =.6004 cos = sin cos 53.1 = sin 36.9 The cosine of an angle is equal to the sine of its complement. First, find the measure of LP using the sine function. Then, find the measure of LP using the cosine function. sine function cosine function L 68o Sine and osine are called co-functions of each other. o-functions of complementary angles are equal. Slide 154 / 240 P 22o 12 M

83 Slide 155 / Given that sin 10 =.1736, write the cosine of a complementary angle. sin 10 =.1736 sin 80 =.9848 cos 10 =.9848 D cos 80 =.1736 Slide 155 () / Given that sin 10 =.1736, write the cosine of sin 10 =.1736 sin 80 =.9848 a complementary angle. D cos 10 =.9848 D cos 80 =.1736 Slide 156 / Given that cos 50 =.6428, write the sine of sin 50 =.7660 sin 40 =.6428 cos 50 =.6428 D cos 40 =.7660 a complementary angle.

84 Slide 156 () / Given that cos 50 =.6428, write the sine of sin 50 =.7660 sin 40 =.6428 a complementary angle. cos 50 =.6428 D cos 40 =.7660 Slide 157 / Given that cos 65 =.4226, write the sine of a complementary angle. sin 25 =.4226 cos 25 =.9063 sin 65 =.9063 D cos 65 = Given that cos 65 =.4226, write the sine of a complementary angle. cos 25 =.9063 sin 65 =.9063 D cos 65 =.4226 sin 25 =.4226 Slide 157 () / 240

85 65 What can you conclude about the sine and cosine of 45 degrees? Slide 158 / 240 Students type their answers here 65 What can you conclude about the sine and cosine of 45 degrees? Slide 158 () / 240 Students type their answers here sin 45 = cos 45 Slide 159 / 240 Solving Right Triangles Return to the Table of ontents

86 Slide 160 / 240 To solve a right triangle means to find all 6 values in a triangle. The lengths of all 3 sides and the measures of all 3 angles. c b a Slide 161 / 240 Let's solve a right triangle given the length of one side and the measure of one acute an gle (S). You need to find the 3 missing parts o z Slide 162 / 240 First, let's find the measure of o y z y

87 Slide 162 () / 240 First, let's find the measure of. 15 y m< + m< = 90o o 64o + m< = 90 64o z m< = 26o Then, let's find the measure of. Slide 163 / o z Then, let's find the measure of o sin64 = y 64 z = y 15 y y y Slide 163 () / 240

88 Slide 164 / 240 Then, let's find the measure of o z Slide 164 () / 240 Then, let's find the measure of a2 + b2 = c2 64o 2 = 15 z 2 z2 + (13.48) z = 225 z2 = z Slide 165 / 240 Try this... Find the missing parts of the triangle. 11 E 37o D R

89 Slide 165 () / 240 Try this... Find the missing parts of the triangle. 11 E 37o D R RD ED m R = 53o Slide 166 / 240 Let's solve a right triangle given the length of two sides (SS). 9 z 15 y Slide 167 / 240 First, find the length of since we know how to do that. ut, how do you find the measure of and? 9 z 15 y

90 First, find the length of since we know how to do that. ut, how do you find the measure of and? 9 Slide 167 () / 240 a2 + b2 = c2 z = 152 z = 225 z z215 = 144 y z = 12 = 12 Slide 168 / 240 You will need to use the inverse trig functions. If sinθ =, θ = sin-1 If cosθ =, θ = cos-1 If tanθ =, θ = tan-1 Pronounced inverse sine, inverse cosine, and inverse tangent. c b θ a With the sine, cosine and tangent trig functions, if you know the angle θ and the measure of one leg, then you can find the measure of a leg of a triangle. With the inverse sine, inverse cosine and inverse tangent trig functions, if you know the measures of 2 legs of a triangle, you can find the measure of the angle. The 3 Inverse Trigonometric Ratios θ = sin-1( opposite side ) hypotenuse θ = tan-1( opposite side ) θ = cos-1( adjacent side ) adjacent side hypotenuse Use the inverse trig function to find the unknown angle measure when you know the length of 2 sides. Remember: c b a θ Slide 169 / 240

91 Slide 170 / Find sin Round the angle measure to the nearest hundredth. Slide 170 () / Find sin Round the angle measure to the nearest hundredth. θ = Slide 171 / Find tan Round the angle measure to the nearest hundredth.

92 Slide 171 () / Find tan Round the angle measure to the nearest hundredth. θ = Slide 172 / Find cos Round the angle measure to the nearest hundredth. 68 Find cos Round the angle measure to the nearest hundredth. θ = Slide 172 () / 240

93 Slide 173 / 240 To find an unknown angle measure in a right triangle, You need to identify the correct trig functionthat will find the missing value. Use SOHHTO to help. Using cosine. θ 15 hyp, I have a and h, so the ratio is a/h which is 9 adj is your angle of reference. Label the two given sides of your triangle opp, adj, or hyp. Identify the trig funtion that uses, and the two sides. now you can solve for, the missing angle using the inverse trig function. How are you going to calculate the measure of? Slide 173 () / 240 To find an unknown angle measure in a right triangle, You need to identify the correct trig functionthat will find the missing value. Use SOHHTO to help. is your angle of reference. Label the two given sides of your triangle opp, adj, or hyp. Identify the trig funtion that uses, and the two sides. Using cosine adj θ y15 hyp, I have a and h, so the ratio is a/h which is now you can solve for, the[this missing object is a pullangle tab] using the inverse trig function. How are you going to calculate the measure of? Slide 174 / 240 Eample Find the trig equation that will find θ. 12 θ 7

94 Slide 174 () / 240 Eample Find the trig equation that will find θ. θ 12 7 θ adj 7 12 opp Slide 175 / 240 Eample Find the trig equation that will find θ. θ Slide 175 () / 240 Eample Find the trig equation that will find θ. 10 θ 12 adj θ 10 hyp 12

95 Slide 176 / 240 Eample Find the trig equation that will find θ. θ 12 9 Slide 176 () / 240 Eample Find the trig equation that will find θ θ θ hyp 12 9 opp Slide 177 / Which is the correct trig equation to solve for 7 12 D E D

96 Slide 177 () / Which is the correct trig equation to solve for 7 12 D E D Slide 178 / Which is the correct trig equation to solve for 12 5 D D E Slide 178 () / Which is the correct trig equation to solve for D D 5 E 12 D

97 Slide 179 / 240 K 11 J D 9 71 Which is the correct trig equation to solve for L Slide 179 () / Which is the correct trig equation to solve for K 11 J D 9 L Slide 180 / 240 Try this... Solve the right triangle. Round your answers to the nearest hundredth. R Q 7 S 24

98 Slide 180 () / 240 Try this... Solve the right triangle. Round your answers to the nearest hundredth. R 24 Q 7 S QS = 25 m Q = 73.74o m S = 16.26o Slide 181 / Find E. 5 E D 8 Slide 181 () / Find E. 5 E Use the Pythagorean Theorem D 8

99 Slide 182 / Find m. 5 E D 8 Slide 182 () / Find m. 5 E Use inverse tangent D 8 Slide 183 / Find the m E. 5 E D 8

100 Slide 183 () / Find the m E. 5 From before, E D 8 Slide 184 / Find the m G. o 20 L 18 G Slide 184 () / Find the m G. o 20 L 18 G

101 Slide 185 / 240 o Find L. L 18 G Slide 185 () / 240 o Find L. L 18 G Slide 186 / Find the m P. P 49.19o o D 56.31o E 18 N 33.69o

102 Slide 186 () / Find the m P. P 49.19o 41.81o D 56.31o 33.69o 12 D E 18 N Slide 187 / Find RT. S T o D 9.53 R Slide 187 () / Find RT. S T D o R

103 Slide 188 / 240 ngle of Elevation and Depression Return to the Table of ontents Slide 189 / 240 How can you use trigonometric ratios to solve word problems involving angles of elevation and depression? Slide 190 / 240 When you look up at an object, the angle your line of sight makes with a line drawn horizontally is the angle of elevation.

104 Slide 191 / 240 When you look down at an object, the angle your line of sight makes with a line drawn horizontally is the angle of depression. Slide 192 / 240 The angle of elevation and the angle of depression are both measured relative to parallel horizontal lines, they are equal in meaure. 79 How can you describe the angle relationship between the angle of elevation and the angle of depression? alternate eterior angles D none of the above corresponding angles alternate interior angles Slide 193 / 240

105 79 How can you describe the angle relationship between the angle of elevation and the angle of depression? Slide 193 () / 240 corresponding angles alternate interior angles alternate eterior angles D none of the above Slide 194 / 240 Eample my is flying a kite at an angle of 58o. The kite's string is 158 feet long and my's arm is 3 feet off the ground. 15 8f ee t 58 o How high is the kite off the ground? 3 feet Slide 195 / f t 58o sinθ = 158 sin58 = = 158 = 134 Now, we must add in my's arm height = 137 The kite is about 137 feet off the ground.

106 Slide 196 / 240 Eample You are standing on a mountain that is 5306 feet high. You look down at your campsite at angle of 30o. If you are 6 feet tall, how far is the base of the mountain from the campsite? 6 ft 30o 5306 ft Slide 197 / 240 tan30 = 5312 ft 30o.5774 = = ,200 ft The campsite is about 9,200 ft from the base of the mountain. Slide 198 / 240 Try this... You are looking at the top of a tree. The angle ofelevation is 55o. The distance from the top of the tree to your position (line of sight) is 84 feet. If you are 5.5 feet tall, how far are you from the base of the tree?

107 Slide 198 () / 240 Try this... You are looking at the top of a tree. The angle ofelevation is 55o. The distance from the top of the tree toftyour position (line of sight) is 84 feet. If you are 5.5 feet tall,84how far are you from the base of the tree? 55o cos55 = 84 = You are approimately 48 ft objectof is athe pull tab] from the[this base tree. Slide 199 / When you look down at an object, the angle your line of sight makes with a line drawn horizontally is the angle of. elevation depression elevation depression 80 When you look down at an object, the angle your line of sight makes with a line drawn horizontally is the angle of. Slide 199 () / 240

108 Slide 200 / Katherine looks down out of the crown of the statue of liberty to an incoming ferry about 345 feet. The distance from crown to the ground is about 250 feet. What is the angle of depression? 81 Katherine looks down out of the crown of the statue of liberty to an incoming ferry about 345 feet. The crown distance from crown to the ground is about 250 feet. What is the angle of depression? 345 ft Slide 200 () / ft ferry The angle of[this depression about object is is a pull tab]46 degrees. Slide 201 / What is the distance from the ferry to the base of the statue?

109 82 What is the distance from thecrown ferry to the base of the statue? 345 ft Slide 201 () / ft base of the statue ferry The ferry is about 238 feet away from the statue. Slide 202 / 240 Law of Sines and Law of osines Return to the Table of ontents How can you solve a non-right triangle? How can you find the side lengths and angle measures of non-right triangles? The Law of Sines and Law of osines can be used to solve any triangle. You can use the Law of Sines when you are given 1. Two angle measures and any side length (S or S) 2. Two side lengths and the measure of a non-included angle (SS) when the angle is a right angle or an obtuse angle. The Law of Sines has a problem dealing with SS when the angle is acute. There can be zero, one or two solutions. lick on: Khan cademy Video "More On Why SS Is Not Postulate" for more info. You can use the Law of osines when you are given 3. Three side lengths (SSS) 4. Two side lengths and the measure of an included angle (SS) Slide 203 / 240

110 Slide 204 / 240 Law of Sines b a c If has sides of length a, b, and c, then sin = sin = sin a b c To use the Law of Sines, 2 angles and 1 side must be given. If has sides of length a, b, and c, then sin = sin = sin a b c Given: has sides of length a, b, and c Prove: sin = sin = sin a b c Slide 205 / 240 Let's prove the Law of Sines b h a c Reasons Statements with side lengths a, b, and c click Given Def of ltitude Draw an altitude from to side click Let h be the length of the altitude Def of sine click Multiply click by b. Mult Prop of =. Multiply click by a. Mult Prop of =. Substitution Prop of = click Divide by ab. Division Prop of = click Prove the Law of Sines (continued) Given: has sides of length a, b, and c Prove: sin = sin = sin a b c b h c Slide 206 / 240 a b g Statements a c Reasons Def of ltitude Draw an altitude from to side click Let g be the length of the altitude Def clickof sine Multiply click by c. Mult Prop of =. Multiply click by a. Mult Prop of =. Substitution Prop of = click Divide by ac. Division Prop of = click Substitution Prop of = click

111 Select the ratios based on the given information. Given: m, m and (side c) (S) 10 65o b Select the ratios based on the given information. 65o b Given: m, m and (side c) (S) 10 70o Slide 207 () / 240 sin = sin = sin a b c Use the Law of Sines to solve the triangle. a Which ratios must be used first? Why? sinmust = sin Which ratios be used first? Why? b c There are 4 numbers in a proportion. If you know 3 of the numbers you can find the 4th. First we can find the length side b. Slide 208 / 240 a 70o o sin = sin b c sin70 = sin65 b 10 b 70o a Slide 207 / 240 sin = sin = sin a b c Use the Law of Sines to solve the triangle.

112 First we can find the length side b. Slide 208 () / 240 a 70o 10 65o b sin = sin.9397 =.9063 b c b 10 sin70 = sin65 b b = b efore we find the length of side a, we find the m. Slide 209 / 240 a 70o 10 65o b=10.37 efore we find the length of side a, we find the m. Triangle Sum Theorem m + m + m = 180o a 70o 65o b=10.37 Triangle Sum Theorem m + m + m = 180o Slide 209 () / 240 m + 70o + 65o = 180o m + 135o = 180o m = 45o 10

113 Slide 210 / 240 Now we find the length side a. a 70o =45o o b=10.37 sin = sin a c a sin = sin a c Slide 210 () / 240 Now we find the length side a. 70o 65o 10 =45o b=10.37 sin45 = sin65 a =.9063 a a = a 7.8 Slide 211 / 240 Try this... Use the Law of Sines to find the length of side b (S). 85o 9 29o a Since the length of the side opposite < is given, find the m< first. hint b

114 Slide 211 () / 240 Try this... Use the Law of Sines to find the length of side b (S). b 85o 9 29o a Since the length of the side opposite < is given, find the m< first. hint b 9.81 Slide 212 / 240 Eample... Find the length of side b (SS with an obtuse angle). 101o b 2.8 Eample... Find the length of side b (SS with an obtuse angle). m< =180 sin = sin m<=58.91 a c sin 8 = sin101 sin = sin 2.8 a b o sin20.09 = sin b sin =.9816 b.3436 = b b = 6.98 sin =.3436 =sin-1(.3436) = Slide 212 () / 240

115 Slide 213 / Find the m. 70o 31o c 10 81o 29o b D 28o 19o Slide 213 () / Find the m. 19o 70o 31o c 10 29o D 28o 81o b Slide 214 / Which ratio must be used to find the length of b or c? 70o c 81o sin a sin b b sin b D sin c 10

116 Slide 214 () / Which ratio must be used to find the length of b or c? c 70o 10 81o b sin a sin b sin D sin b c Slide 215 / What is the length of b? c 70o 81o 10 b Slide 215 () / What is the length of b? c 10 81o 70o b

117 Slide 216 / What is the length of c? c 70o 10 81o b Slide 216 () / What is the length of c? c 10 81o 70o b Slide 217 / 240 Law of osines c a b If has sides of length a, b, and c, then: To use the Law osines, you must be given the length of 3 sides (SSS) or the length of 2 sides and the measure of the included angle (SS).

118 If has sides of length a, b, and c, then Given: b a b c h a D c- c Reasons Statements has sides of length a, b, and c Slide 218 / 240 Let's prove the Law of osines with side lengths a, b, and c Given click Def of ltitude Draw an altitude D from to side click. Let h be the length of the alt. Prove: (similar reasoning shows that ) Let be the length of D. Then (c-) is the length of D. Segment ddition Postulate click In Definition of cosine click D, cos = /b (1) =b(cos) Multiply click by b. Mult Prop of =. (2) In Pythagorean Theorem click In D, D, Pythagorean Theorem click Simplify click Substitution, equation (2) click ssociative Prop of ddition click Substitution, equation (1) click Eample Use the Law of osines to solve the right triangle. Slide 219 / 240 a=16 a is opposite < b is opposite < c is opposite < c=27 b=23 The formula you choose depends on which angle you are solving for first. Slide 220 / 240 c=27 a=16 To find the m, b=23 a2 = b2 + c2-2bc(cos) 162 = (23)(27)(cos) 256 = (cos) 256 = (cos) = -1242(cos).8068 = cos = cos-1(.8068) m 36.22o

119 Slide 221 / 240 c=27 a= To find the m, b=23 b2 = a2 + c2-2ac(cos) 232 = (16)(27)(cos) 529 = (cos) 579 = (cos) -406 = -864(cos).4699 = cos =cos-1(.4699) m 61.97o or Using 2 different methods, the answers are slightly different because of rounding. Slide 222 / 240 c= b=23 To find the m, Use the Triangle Sum Theorem. Slide 222 () / 240 a= c= b=23 m + m + m = 180o 36.22o o + m = 180o 98.19o + m = 180o Use the Triangle Sum Theorem. m 81.81o To find the m, a=16

120 Slide 223 / 240 Try this... Use the Law of osines to find the m< (SSS) Slide 223 () / 240 Try this... Use the Law of osines to find the m< (SSS) b2 = a2 + c2-2ac(cos) 72 = (5)(6)(cos) 49 = (cos) 49 = 61-60cos -12 = -60cos.2 = cos m< 78.46o Slide 224 / In the triangle the length of c is

121 Slide 224 () / In the triangle the length of c is Slide 225 / In the triangle the length of a is Slide 225 () / In the triangle the length of a is

122 Slide 226 / Which formula would you use to find the m<? a2 = b2 + c2-2ac(cos) a2 = b2 + c2-2bc(cos) b2 = a2 + c2-2ac(cos) D c2 = a2 + b2-2ab(cos) Slide 226 () / Which formula would you use to find the m<? a2 = b2 + c2-2ac(cos) a2 = b2 + c2-2bc(cos) b2 = a2 + c2-2ac(cos) D c2 = a2 + b2-2ab(cos) Slide 227 / What is the m?

123 Slide 227 () / What is the m? a2 = b2 + c2-2bc(cos) 82 = (9)(15)(cos) 64 = (cos) 64 = cos -242 = -270cos.8963 = cos m 26.32o Slide 228 / What is the m? Slide 228 () / What is the m? c2 = a2 + b2-2ab(cos) 152 = (8)(9)(cos) 225 = (cos) 225 = cos 80 = -144cos = cos =cos-1(-.5556) m o

124 Slide 229 / What is the measure of (S)? Students type their answers here Slide 229 () / What is the measure of (S)? 8 Students type their answers here a2 = b2 + c2-2bc(cos50) a2 = (8)(4)(.6428) 4 a2 = a2 = a = 6.23 or b2 = a2 + c2-2ac(cos) 82 = (6.23)(4)(cos) 64 = (cos) 49 = cos = cos.1166 = cos m 83.3o Slide 230 / The Law of Sines and osines is used to solve... acute triangles obtuse triangles D all triangles right triangles

125 93 The Law of Sines and osines is used to solve... Slide 230 () / 240 right triangles acute triangles D obtuse triangles D all triangles Slide 231 / 240 rea of an Oblique Triangle Return to the Table of ontents Do you remember this? Previously, we found the area of a triangle when we were given 3 sides. Find the area of the triangle. 13 feet 13 feet 10 feet Slide 232 / 240

126 = 1 2 bh Slide 233 / 240 b is the base of the triangle b = 10. h is the altitude (or height). It is the perpendicular bisector of the base in an isosceles triangle. Find h, using the pythagorean theorem - 13 feet 13 feet h 5 feet 5 feet What formula can you use to find the area of a triangle if you know the length of two sides and the measure of an included angle (SS)? Slide 234 / 240 Find the area of the triangle. 13 feet feet Slide 235 / 240 Since = 1 bh 2 and b = 10, we need to find h. 13 feet h feet

127 Slide 236 / 240 Let's derive the formula for an oblique triangle. Given: has sides of lengtha, b, and c. ltitude h. Prove: a h Reasons Statements click with side lengths a, b, and c Given Draw an altitude from to side Def clickof ltitude c Let h be the length of the altitude b Def clickof sine Multiply click by a. Mult Prop of =. Definition. Formula for the area of a triangle. click Substitution Prop of = click ommutative Prop of Multiplication click Slide 237 / 240 D E F 94 Which of the following epressions can be used to find the area of the triangle below? Select all that apply. Slide 237 () / Which of the following epressions can be used to find the area of the triangle below? Select all that apply. D,, E E F

128 Slide 238 / Find the area of the triangle to the nearest tenth. Students type their answers here Slide 238 () / Find the area of the triangle to the nearest tenth. Students type their answers here Slide 239 / Find the area of the triangle to the nearest tenth. Students type their answers here

129 Slide 239 () / Find the area of the triangle to the nearest tenth. Students type their answers here Slide 240 / Find the area of the triangle to the nearest tenth. Students type their answers here 97 Find the area of the triangle to the nearest tenth. Students type their answers here or Slide 240 () / 240