New Jersey enter for Teaching and Learning Slide 1 / 240 Progressive Mathematics Initiative This material is made freely available at www.njctl.org 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: www.njctl.org Geometry Slide 2 / 240 Trigonometry of Right Triangles 2014-06-05 www.njctl.org Table of ontents Pythagorean Theorem Similarity in Right Triangles Special Right Triangles Trigonometric Ratios Solving Right Triangles ngles of Elevation and epression Law of Sines and Law of osines rea of an Oblique Triangle lick on a Topic to go to that section Slide 3 / 240
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. leg hypotenuse 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.
Pythagorean Theorem Slide 7 / 240 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. leg 2 + leg 2 = hypotenuse 2 or a 2 + b 2 = c 2 a b c Slide 8 / 240 Eample: Find the length of the missing side of the right triangle. 9 12 Is the missing side a leg or the hypotenuse of the right triangle? Slide 8 (nswer) / 240 Eample: Find the length of the missing side of the right triangle. nswer hypotenuse 9 = 15 12 Is the missing side a leg or the hypotenuse [This object is a pull of tab] the right triangle?
Solve for : Slide 9 / 240 9 2 + 12 2 = 2 81 + 144 = 2 225 = 2 # 15 = -15 is a etraneous solution, a distance can not equal a negative number. = 15 12 9 Eample: Find the length of the missing side of the right triangle. Is the missing side a leg or the hypotenuse of the right triangle? 28 20 Slide 10 / 240 Eample: Slide 10 (nswer) / 240 Find the length of the missing side of the right triangle. Is the missing side a leg or the hypotenuse of the right triangle? nswer leg 28 20
1 The missing side is the of the right triangle. Slide 11 / 240 leg hypotenuse 6 9 1 The missing side is the of the right triangle. Slide 11 (nswer) / 240 leg hypotenuse nswer 6 9 2 Find the length of the missing side. Slide 12 / 240 6 9
2 Find the length of the missing side. Slide 12 (nswer) / 240 6 9 nswer 3 The missing side is the of the right triangle. Slide 13 / 240 leg hypotenuse 15 36 3 The missing side is the of the right triangle. Slide 13 (nswer) / 240 leg hypotenuse nswer 36 15
4 Find the length of the missing side. Slide 14 / 240 15 36 4 Find the length of the missing side. Slide 14 (nswer) / 240 nswer 36 15 Real World pplication Slide 15 / 240 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
Slide 16 / 240 2 2 2 Solve using a + b = c? 28 feet 7 feet The ladder will reach feet up the wall safely. Slide 16 (nswer) / 240 2 2 2 Solve using a + b = c nswer? 7 feet 28 feet The ladder will reach feet up the wall safely. Real World pplication 84 Slide 17 / 240 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?
Real World pplication 84 Slide 17 (nswer) / 240 nswer 50 84 2 + 50 2 = 2 9556 = 2 97.75 = The court is 97.75 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? 5 N court is 50 feet wide and the length from one corner of the court to the opposite corner is 106.5 feet. How long is the court? (Round the answer to the nearest whole number) Slide 18 / 240 94.03 feet 117.7 feet 118 feet 94 feet 5 N court is 50 feet wide and the length from one corner of the court to the opposite corner is 106.5 feet. How long is the court? (Round the answer to the nearest whole number) Slide 18 (nswer) / 240 94.03 feet 117.7 feet 118 feet 94 feet nswer
Pythagorean Theorem pplications Slide 19 / 240 The Pythagorean Theorem can also be used in figures that contain right angles. Eample Find the perimeter of the square. Slide 20 / 240 18 cm Psq = 4s note: efore finding the perimeter of the square, we need to first find the length of each side. Remember, in a square all sides are congruent. Slide 21 / 240 18 cm Start here: 2 + 2 = 18 2
Remember, in a square all sides are congruent. Slide 21 (nswer) / 240 18 cm Start here: 2 + 2 = 18 2 nswer 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 Slide 22 / 240 10 feet y definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. Slide 23 / 240 13 feet h 13 feet 5 feet 5 feet
y definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. Slide 23 (nswer) / 240 13 feet h 13 feet nswer 5 feet 5 feet Try this... Find the perimeter of the rectangle. Slide 24 / 240 8 in Prect = 2l + 2w 10 in Try this... Find the perimeter of the rectangle. Slide 24 (nswer) / 240 8 in 10 in nswer Prect = 2l + 2w 2 + 8 2 = 10 2 = 6 P rect = 2(6) + 2(8) P = 28 inches
6 Find the area of the rectangle. Slide 25 / 240 120 square feet 84 square feet 46 square inches 8 feet 17 feet 46 square feet 6 Find the area of the rectangle. Slide 25 (nswer) / 240 120 square feet 84 square feet nswer 46 square inches 8 feet 17 feet 46 square feet 7 Find the perimeter of the square. (Round to the nearest tenth) Slide 26 / 240 12.8 cm 25.5 cm 25.6 cm 36 cm 9 cm
7 Find the perimeter of the square. (Round to the nearest tenth) Slide 26 (nswer) / 240 12.8 cm 25.5 cm 9 cm 25.6 cm 36 cm nswer 8 Find the area of the triangle. Slide 27 / 240 7 inches 7 inches 10 inches 8 Find the area of the triangle. Slide 27 (nswer) / 240 nswer 7 inches 10 inches 7 inches h=4.9in =24.5 square inches
9 Find the area of the triangle. Slide 28 / 240 7 inches 7 inches 4 inches 9 Find the area of the triangle. Slide 28 (nswer) / 240 7 inches 7 inches nswer 4 inches h=6.7in =13.4 square inches onverse of the Pythagorean Theorem Slide 29 / 240 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 2 = a 2 + b 2, then is a right triangle. a c b
Eample Tell whether the triangle is a right triangle. 24 E Slide 30 / 240 Remember c is the longest side 25 F 7 nswer Eample Tell whether the triangle is a right triangle. If c 2 = a 2 + b 2, then If then 24 is a right triangle. is a right triangle. Remember c is the longest therefore side is a right triangle. 25 F E 7 Slide 30 (nswer) / 240 Theorem Slide 31 / 240 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 2 > a 2 + b 2, then is obtuse. a c b
Theorem Slide 32 / 240 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. c a If c 2 < a 2 + b 2, then is acute. b Eample lassify the triangle as acute, right, or obtuse. Slide 33 / 240 15 13 17 Eample lassify the triangle as acute, right, or obtuse. Slide 33 (nswer) / 240 c = 17 15 13 nswer Since the triangle is acute. 17
10 lassify the triangle as acute, right, obtuse, or not a triangle. Slide 34 / 240 acute right 12 obtuse 15 not a triangle 11 10 lassify the triangle as acute, right, obtuse, or not a triangle. Slide 34 (nswer) / 240 acute right obtuse not a triangle nswer 15 12 11 11 lassify the triangle as acute, right, obtuse, or not a triangle. Slide 35 / 240 acute right obtuse 3 5 not a triangle 6
11 lassify the triangle as acute, right, obtuse, or not a triangle. Slide 35 (nswer) / 240 acute right obtuse nswer 5 3 not a triangle 6 12 lassify the triangle as acute, right, obtuse, or not a triangle. Slide 36 / 240 acute right 25 obtuse not a triangle 20 19 12 lassify the triangle as acute, right, obtuse, or not a triangle. Slide 36 (nswer) / 240 acute right obtuse not a triangle nswer 20 25 19
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 (nswer) / 240 acute right obtuse nswer 14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. Slide 38 / 240 acute triangle right triangle obtuse triangle
14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. Slide 38 (nswer) / 240 acute triangle right triangle nswer obtuse triangle Slide 39 / 240 Review If c 2 = a 2 + b 2, then triangle is right. If c 2 > a 2 + b 2, then triangle is obtuse. If c 2 < a 2 + b 2, then triangle is acute. Slide 40 / 240 Similarity in Right Triangles Return to the Table of ontents
There are many proofs to the Pythagorean Theorem. How many do you know? Slide 41 / 240 Triangle similarity can be used to prove the Pythagorean Theorem. How? Theorem The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. Slide 42 / 240 is the altitude of ~ ~ To prove this, click for Lab 1 - Similar Right Triangles Slide 43 / 240 Therefore, the altitude of a right triangle divides the triangle into two smaller triangles that are similar click to the original triangle and similar click to each other.
To prove this, click for Lab 1 - Similar Right Triangles See Similar Right Triangles Lab to complete this activity. Slide 43 (nswer) / 240 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? Therefore, the altitude of [This a right object is triangle a pull tab] divides the triangle into two smaller triangles that are similar click to the original triangle and similar click to each other. Let's prove the Theorem. Slide 44 / 240 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: is a right triangle is the altitude of ~ ~ Statements is a right triangle is a right angle is a right angle ~ is a right angle ~ ~ ~ Reasons click Given click Given ef clickof ltitude ef of Perp Lines. 2 lines that form a rt click angle click ll rt angles are click Refleive Prop of click ~ click ef of Perp Lines click ll rt angles are click Refleive Prop of click ~ click Transitive Prop of ~ Let's sketch the 3 triangle's separately, with the same orientation. Slide 45 / 240 Match up the angles. Helpful tip: If you set, then you can assign all the angles a value and easily find the matches. 30 30 30 30 60 30 60 60 60 60
c d b e ssign lengths to all the segments. Let the lengths of the segments on the hypotenuse be d and e. a c a a b e b d Label the sides of a triangle with the lower case letter of the opposite angle. ~ ~ Slide 46 / 240 ecause the triangles are similar the corresponding sides are proportional. ~ ~ To prove the Pythagorean Theorem, use the proportions. Given: Prove: is a right triangle. is an altitude. Statements ~ Reasons ltitude of a rt triangle theorem. click efinition of similar triangles. click Using the multiplication property of equality, multiply click the equation by bc. Slide 47 / 240 d c b e a (1) ~ simplify click ltitude of a rt triangle theorem. click efinition of similar triangles. click Using the multiplication property of equality, click multiply the equation by ac (2) click simplify To prove the Pythagorean Theorem, use the proportions (continued). Given: Prove: is a right triangle. is an altitude. Statements Reasons Using the addition property of equality, add equation (1) click and equation (2) together. click istributive Property click Given click Substitution Slide 48 / 240 c e a click Simplify d b
Eample Slide 49 / 240 Find the length of the altitude KI? H 12 I 13 K 5 J 12 H It maybe helpful to sketch the 3 triangle's separately, with the same orientation. I 13 5 K 12 H 13 H 12 Slide 50 / 240 K 5 J J I ecause the triangles are similar the corresponding sides are proportional. K I J 5 K 13 = 60 4.62 Try this... Find the length of RS. P 3 Q R 5 4 Slide 51 / 240 S S R R 5 4 P 3 Q S 4 Q P 5 S
Try this... Find the length of RS. P 3 Q R 5 4 Slide 51 (nswer) / 240 S 5 S 4 R nswer R P 3 Q S 4 Q P 5 S 15 Which ratio is the ratio of corresponding sides? Slide 52 / 240 I H K J 15 Which ratio is the ratio of corresponding sides? Slide 52 (nswer) / 240 I H nswer K J
16 Find KJ. Slide 53 / 240 I 7 24 H K 25 J 16 Find KJ. Slide 53 (nswer) / 240 I 7 24 Set KJ = H nswer K 25 J The net two theorems are Geometric Mean Theorems. Slide 54 / 240 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
The geometric mean of two positive numbers a and b is the positive number that satisfies a = b 2 = ab = Slide 55 / 240 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. Eample Slide 56 / 240 Find the geometric mean of 8 and 14. 2 = 8(14) 2 = 112 (only the positive value) 17 Find the geometric mean of 7 and 56. Write the answer is simplest radical form. Slide 57 / 240
17 Find the geometric mean of 7 and 56. Write the answer is simplest radical form. Slide 57 (nswer) / 240 nswer 18 Find the geometric mean of 3 and 48. Slide 58 / 240 Students type their answers here 18 Find the geometric mean of 3 and 48. Slide 58 (nswer) / 240 Students type their answers here nswer
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. Slide 59 / 240 is the altitude of Since, ~ 2 = () Eample Slide 60 / 240 Find z. 8 6 z Eample Slide 60 (nswer) / 240 Find z. 8 6 z nswer
Eample Slide 61 / 240 Find z. 18 z 6 Slide 61 (nswer) / 240 Try this... Find y. Slide 62 / 240 1) 18 8 y 2) y 9 12
Try this... Find y. Slide 62 (nswer) / 240 1) 18 8 y 2) y 9 12 nswer 1) y = 12 2) y = 6.75 19 Find. Slide 63 / 240 10 100 20 50 5 19 Find. Slide 63 (nswer) / 240 10 100 5 20 50 nswer
20 Find. Slide 64 / 240 99 11 9 20 Find. Slide 64 (nswer) / 240 99 11 9 nswer 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. Slide 65 / 240 is the altitude of Since, ~ ~ ~ ~ = =
Eample Find. R 4 S 9 T Slide 66 / 240 U Eample Find. R 4 S 9 T Slide 66 (nswer) / 240 U nswer Eample Find. E 4 F Slide 67 / 240 6 G
Eample Find. E 4 F Slide 67 (nswer) / 240 6 G nswer 21 Is PR a geometric mean between QR and SR? True P Slide 68 / 240 False Q S R 21 Is PR a geometric mean between QR and SR? True P Slide 68 (nswer) / 240 False nswer S Q R False
22 Is the geometric mean correct? True P Slide 69 / 240 False Q S R 22 Is the geometric mean correct? True P Slide 69 (nswer) / 240 False nswer S Q R True 23 Which proportion is correct? Slide 70 / 240 J K L M
23 Which proportion is correct? Slide 70 (nswer) / 240 J K L nswer M 24 Find y. Slide 71 / 240 20 5 12 9 16 y 24 Find y. Slide 71 (nswer) / 240 20 5 nswer 9 16 y 12
25 Find y. Slide 72 / 240 3 18 24 None of the above 27 y 9 25 Find y. Slide 72 (nswer) / 240 3 18 24 nswer None of the above 27 y 9 26 Find. Slide 73 / 240 5 8
26 Find. Slide 73 (nswer) / 240 nswer 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. Slide 75 / 240 45-45-90 30-60-90 45 o 60 o 90 o 30 o 90 o 45 o
45-45-90 Triangle Theorem Slide 76 / 240 45-45-90 triangle is an isosceles right triangle, where the hypotenuse is 2 times the length of the leg. hypotenuse = leg( 2) an you prove this? 45 o 2 45 o 45-45-90 Triangle Theorem Slide 76 (nswer) / 240 45-45-90 triangle is an isosceles right triangle, where the hypotenuse is 2 times the length of the leg. hypotenuse = leg( 2) an you prove this? nswer 45 o 2 45 o Eample Find the length of the missing sides. Write the answer in simplest radical form. P 45 o 6 Q Slide 77 / 240 y 45 o R
Eample Find the length of the missing sides. Write the answer in simplest radical form. y the orollary 6 P to the ase ngles Thm, 45 o PQ=QR. y=6 Q Slide 77 (nswer) / 240 nswer hypotenuse = 2(leg) = 2(6) = 6 2 y 45 o R Eample Find the length of the missing sides of the right triangle. S y T Slide 78 / 240 18 V Eample Find the length of the missing sides of the right triangle. Since, STU is an hypotenuse = leg S y isosceles rt triangle nswer ST=TV =y There are 2 ways to solve. 18 T Slide 78 (nswer) / 240 V
Try this... Find the length of the missing sides. Slide 79 / 240 y 8 Try this... Find the length of the missing sides. Slide 79 (nswer) / 240 y nswer 8 27 Find the value of. Slide 80 / 240 5 5# 2 (5# 2)/2 y 5
27 Find the value of. Slide 80 (nswer) / 240 5 5# 2 (5# 2)/2 nswer y 5 28 Find the value of y. 5 Slide 81 / 240 5# 2 (5# 2)/2 y 5 28 Find the value of y. 5 Slide 81 (nswer) / 240 5# 2 (5# 2)/2 nswer y 5
29 What is the length of the hypotenuse of an isosceles right triangle, if the length of the legs is 8# 2 inches. Slide 82 / 240 29 What is the length of the hypotenuse of an isosceles right triangle, if the length of the legs is 8# 2 inches. Slide 82 (nswer) / 240 hypotenuse = leg( ) nswer 30 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm. Slide 83 / 240
30 What is the length of each leg of an isosceles, if the length of the hypotenuse hypotenuse is 20 cm. = leg( ) Slide 83 (nswer) / 240 nswer 30-60-90 Triangle Theorem Slide 84 / 240 In a 30-60-90 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. 60 o # 3 2 30 o hypotenuse = 2(shorter leg) longer leg = 3(shorter leg) This can be proved using an equilateral triangle. 60 o 2 Slide 85 / 240 For right triangle, is a perpendicular bisector. let a =, c = 2 and b= # 3 30 o c=2 30 30 b 2 60 60 a=
Eample Find the length of the missing sides of the right triangle. G 30 o Slide 86 / 240 y H 5 60 o F Recall triangle inequality, the shortest side is opposite the smallest angle and the longest side is opposite the largest angle. G 30 o Slide 87 / 240 HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF y H 5 60 o F Recall triangle inequality, the shortest side is opposite the hypotenuse smallest angle = 2(shorter and the leg) longest side is opposite the largest angle. = 2(5) HF is the shortest side = 10 GF is the longest side (hypotenuse) GH is the 2nd longer longest leg = side # 3(shorter leg) HF < GH < GF y = # 3(5) nswer y = 5 # 3 y H G 30 o 5 60 o F Slide 87 (nswer) / 240
Eample Find the length of the missing sides of the right triangle. M 60 o Slide 88 / 240 9 y 30 o T Eample Find the length of the missing sides of the right triangle. M 60 o Slide 88 (nswer) / 240 nswer M is the shorter leg and MT is the longer leg 9 longer leg = # 3(shorter leg) 9 = # 3() 3# 3 = 30 o y T Slide 89 / 240 Eample Find the area of the triangle. 14 ft
The altitude (or height) divides the triangle into two 30 o -60 o -90 o triangles. h 14 ft Slide 90 / 240?? The length of the shorter leg is 7 ft. The length of the longer leg is 7 3 ft. = b(h) = 14(7 # 3) # 84.87 square ft Try this... Find the length of the missing sides of the right triangle. 15 30 o y 60 o Slide 91 / 240 Try this... Find the length of the missing sides of the right triangle. 15 nswer 30 o y 60 o Slide 91 (nswer) / 240
Try this... Find the area of the triangle. 9 ft Slide 92 / 240 30 o Try this... Find the area of the triangle. 9 ft Slide 92 (nswer) / 240 short leg = 4.5 ft long leg = 4.5 ft 30 o nswer 31 Find the value of. Slide 93 / 240 7 7# 3 (7# 2)/2 7 60 o 30 o 14
31 Find the value of. Slide 93 (nswer) / 240 7 7# 3 (7# 2)/2 nswer 7 60 o 30 o 14 32 Find the value of. Slide 94 / 240 7 7# 3 (7# 2)/2 14 7# 2 32 Find the value of. Slide 94 (nswer) / 240 7 7# 3 (7# 2)/2 nswer 7# 2 14
33 Find the value of. 7 7# 3 (7# 2)/2 30 o 7 3 60 o Slide 95 / 240 14 33 Find the value of. 7 7# 3 (7# 2)/2 14 nswer 30 o 7 3 60 o Slide 95 (nswer) / 240 34 The hypotenuse of a 30 o -60 o -90 o triangle is 13 cm. What is the length of the shorter leg? Slide 96 / 240
34 The hypotenuse of a 30 o -60 o -90 o triangle is 13 cm. What is the length of the shorter leg? Slide 96 (nswer) / 240 nswer shorter leg = 13/2 shorter leg = 6.5cm 35 The length the longer leg of a 30 o -60 o -90 o triangle is 7 cm. What is the length of the hypotenuse? Slide 97 / 240 35 The length the longer leg of a 30 o -60 o -90 o triangle is 7 cm. What is the length of the hypotenuse? Slide 97 (nswer) / 240 nswer shorter leg = hypotenuse = =
Real World Eample Slide 98 / 240 The wheelchair ramp at your school has a height of 2.5 feet and rises at angle of 30 o. What is the length of the ramp? Slide 99 / 240? 2.5 30 o The triangle formed by the ramp is a 30 o -60 o -90 o 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. 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. Slide 100 / 240 45 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. Slide 100 (nswer) / 240 nswer? 45 o 3 feet 37 What is the length of the base of the ramp? Round to the nearest hundredth. Slide 101 / 240 45 o 3 feet? 37 What is the length of the base of the ramp? Round to the nearest hundredth. Slide 101 (nswer) / 240 nswer 45 o 3 feet?
38 The yield sign is shaped like an equilateral triangle. Find the length of the altitude. Slide 102 / 240 20 inches Slide 102 (nswer) / 240 39 The yield sign is shaped like an equilateral triangle. Find the area of the sign. Slide 103 / 240 20 inches
39 The yield sign is shaped like an equilateral triangle. Find the area of the sign. Slide 103 (nswer) / 240 20 inches nswer 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. b c a
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 Engineers can measure the angle of slant using any of the right triangles constructed below. Slide 107 / 240 Engineers very carefully measure the perpendicular distance from a tower window (points, or F) to the ground (points G, E or ). Then they measure the distance from the tower to points, E or G. ~ E~ FG WHY? F GE angle of slant Engineers can measure the angle of slant using any of the right triangles constructed below. Slide 107 (nswer) / 240 Engineers very carefully measure the perpendicular distance from a tower window (points, or F) to the ground (points G, E or ). Then they measure the distance ~ from the tower to points, E or G. ~ E~ FG WHY? nswer F GE angle of slant
Let's calculate the ratio's of the height to the base for each right triangle. Triangle Height ase Ratio Height / ase =50m =5m 50/5=10 E E=30m E=3m 30/3=10 FG FG=20m G=2m 20/2=10 nswer Slide 108 / 240 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. 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 109 / 240 To learn right triangle trigonometry, first you need to be able to identify the sides of a right triangle. Label the sides of a triangle with the lower case letter of the opposite angle. Slide 110 / 240 In a right triangle, there are 2 acute angles. In the triangle to the left, and are the acute angles. b c a
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 hyp Slide 111 / 240 a opp b opp adj a c hyp When is the reference angle, the side opposite is b. the side adjacent (or net to) is a. and the hypotenuse is c. 40 What is the side opposite to J? Slide 112 / 240 JL LK J L KJ K 40 What is the side opposite to J? Slide 112 (nswer) / 240 JL LK J L KJ nswer K
41 What is the hypotenuse of the triangle? Slide 113 / 240 JL J L LK KJ K 41 What is the hypotenuse of the triangle? Slide 113 (nswer) / 240 JL J L LK KJ nswer K 42 What is the side adjacent to J? Slide 114 / 240 JL J L LK KJ K
42 What is the side adjacent to J? Slide 114 (nswer) / 240 JL J L LK KJ nswer K 43 What is the side opposite K? JL LK KJ J L K Slide 115 / 240 43 What is the side opposite K? JL J L Slide 115 (nswer) / 240 LK KJ nswer K
44 What is the side adjacent to K? JL LK KJ J L K Slide 116 / 240 44 What is the side adjacent to K? JL J L Slide 116 (nswer) / 240 LK KJ nswer K Trigonometric Ratios Slide 117 / 240 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). b a c
The 3 Trigonometric Ratios Slide 118 / 240 sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ = opposite side adjacent side This spells... SOHHTO or b θ c 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". Eample Find the sin F, cos F, and tan F. 6 10 Slide 120 / 240 E 8 F Since F is your reference angle, label the sides of the triangle opposite, adjacent and hypotenuse. Use the pneumonic to find the trig ratios. opp 6 E adj 8 10 hyp F sinf = opp 6 hyp = 10 = 3 5 cosf = adj = 8 4 = hyp 10 5 tan F = opp 6 3 = = adj 8 4 lways reduce fractions to lowest terms.
Eample Find the sin, cos, and tan. 6 10 Slide 121 / 240 E 8 F Since is your reference angle, label the sides of the triangle opposite, adjacent and hypotenuse. Use the pneumonic to find the trig ratios. adj 6 E opp 8 10 hyp F sin = opp 8 4 = hyp 10 = 5 cos = adj = 6 3 = hyp 10 5 tan = opp adj = 8 6 lways reduce fractions to lowest terms. 4 = 3 45 What is the sin R? Slide 122 / 240 20/29 21/20 21/29 20/21 45 What is the sin R? Slide 122 (nswer) / 240 20/29 21/20 21/29 20/21 nswer
46 What is the cosr? Slide 123 / 240 20/29 21/20 21/29 20/21 46 What is the cosr? Slide 123 (nswer) / 240 20/29 21/20 21/29 20/21 nswer 47 What is the tanr? Slide 124 / 240 20/21 21/20 20/29 21/29
47 What is the tanr? Slide 124 (nswer) / 240 20/21 21/20 20/29 21/29 nswer 48 What is the sinq? Slide 125 / 240 20/29 21/20 21/29 29/20 48 What is the sinq? Slide 125 (nswer) / 240 20/29 21/20 21/29 29/20 nswer
49 What is the cosq? Slide 126 / 240 20/29 21/20 21/29 29/21 49 What is the cosq? Slide 126 (nswer) / 240 20/29 21/20 21/29 29/21 nswer 50 What is the tanq? Slide 127 / 240 20/29 21/20 21/29 20/21
50 What is the tanq? Slide 127 (nswer) / 240 20/29 21/20 21/29 20/21 nswer The angle of slant of the Tower of Pisa is 84.3 Slide 128 / 240 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). Find the following: sin 84.3 =.9951 click cos 84.3 =.0993 click tan 84.3 = 10.0187 click F angle of slant 51 Evaluate sin 60. Round to the nearest ten thousandth. Slide 129 / 240 0.5 0.8660 1.7321 0.5774
51 Evaluate sin 60. Round to the nearest ten thousandth. Slide 129 (nswer) / 240 0.5 0.8660 1.7321 0.5774 nswer 52 Evaluate cos 60. Round to the nearest ten thousandth. 0.5 Slide 130 / 240 0.8660 1.7321 0.5774 52 Evaluate cos 60. Round to the nearest ten thousandth. 0.5 Slide 130 (nswer) / 240 0.8660 1.7321 0.5774 nswer
53 Evaluate tan 60. Round to the nearest ten thousandth. 0.5 Slide 131 / 240 0.8660 1.7321 0.5774 53 Evaluate tan 60. Round to the nearest ten thousandth. 0.5 Slide 131 (nswer) / 240 0.8660 1.7321 0.5774 nswer Trig tables were used by early mathematicians and astronomers to calculate distances that they were unable to measure directly. Slide 132 / 240 Today, calculators are usually used.
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 Eample Find the trig equation that will find. Slide 134 / 240 30 o 12 Eample Find the trig equation that will find. Slide 134 (nswer) / 240 30 o 12 nswer adj 30 o 12 opp
Eample Find the trig equation that will find. Slide 135 / 240 30 o 12 Eample Find the trig equation that will find. Slide 135 (nswer) / 240 30 o 12 nswer adj 30 o hyp 12 Eample Find the trig equation that will find. Slide 136 / 240 30 o 12
Eample Find the trig equation that will find. Slide 136 (nswer) / 240 30 o 12 nswer 30 o hyp 12 opp 54 Using, which is the correct trig equation needed to solve for. sin40 = 12/ cos40 = /12 tan40 = 12/ sin40 = /12 E 40 o 12 Slide 137 / 240 54 Using, which is the correct trig equation needed to solve for. sin40 = 12/ cos40 = /12 tan40 = 12/ sin40 = /12 nswer E 40 o 12 Slide 137 (nswer) / 240
55 Using, which is the correct trig equation needed to solve for. sin50 = 12/ cos50 = /12 tan50 = 12/ 12 sin50 = /12 E 50 o Slide 138 / 240 55 Using, which is the correct trig equation needed to solve for. sin50 = 12/ cos50 = /12 tan50 = 12/ sin50 = /12 nswer 12 50 o E Slide 138 (nswer) / 240 56 Using J, which is the correct trig equation needed to solve for. tan32 = /11 cos32 = /11 tan32 = 11/ sin32 = 11/ J 32 o K 11 L Slide 139 / 240
56 Using J, which is the correct trig equation needed to solve for. tan32 = /11 cos32 = /11 tan32 = 11/ sin32 = 11/ nswer J 32 o K 11 L Slide 139 (nswer) / 240 57 Using K, which is the correct trig equation needed to solve for. K Slide 140 / 240 tan58 = /11 cos58 = /11 58 o 11 tan58 = 11/ sin 58 = 11/ J L 57 Using K, which is the correct trig equation needed to solve for. K Slide 140 (nswer) / 240 tan58 = /11 cos58 = /11 tan58 = 11/ sin 58 = 11/ nswer J 58 o 11 L
Finding the Missing Side of a Right Triangle Slide 141 / 240 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 10.0187 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 Finding the Missing Side of a Right Triangle Slide 142 / 240 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 Eample Find. Round your answer to the nearest hundredth. Slide 143 / 240 G 25 o E 12 M
Eample Find. Round your answer to the nearest hundredth. G 25 o 12 E nswer sin G = sin25 = EM GM 12 12 (12).4226 = (12) Slide 143 (nswer) / 240 5.07 M Eample Find. Round your answer to the nearest hundredth. Slide 144 / 240 G E 12 65 o M Eample Find. Round your answer to the nearest hundredth. Slide 144 (nswer) / 240 G E cos M = EM GM 12 nswer cos 65 = (12).4226 = 12 12 (12) 65 o 5.07 M
Eample Find y. Round your answer to the nearest hundredth. Slide 145 / 240 10 20 o E y Eample Find y. Round your answer to the nearest hundredth. E tan = E Slide 145 (nswer) / 240 10 20 o nswer tan 20 = (y).3640 = 10 y 10 y (y) E y.3640y = 10 y 27.47 58 Find the length of LM. Round your answer to the nearest tenth. P Slide 146 / 240 12 L 68 o M
58 Find the length of LM. Round your answer to the nearest tenth. P Slide 146 (nswer) / 240 nswer 12 L 68 o M 59 Find the length of LP. Round your answer to the nearest tenth. P Slide 147 / 240 12 L 68 o M 59 Find the length of LP. Round your answer to the nearest tenth. P Slide 147 (nswer) / 240 nswer 12 L 68 o M
Eplain and use the relationship between the sine and cosine of complementary angles. Slide 148 / 240 Slide 149 / 240 Find the measure of? Slide 150 / 240
60 For right triangle, what is the measure of? 30 degrees Slide 151 / 240 50 degrees 60 degrees cannot be determined 30 o 60 For right triangle, what is the measure of? 30 degrees Slide 151 (nswer) / 240 50 degrees 60 degrees nswer cannot be determined [This 30 o object is a pull tab] 61 If the, find the complementary angle? Slide 152 / 240 20 degrees 70 degrees 160 degrees none of the above
61 If the, find the complementary angle? Slide 152 (nswer) / 240 20 degrees 70 degrees 160 degrees nswer 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 53.1 + 36.9 = 90 36.9 Slide 153 / 240 sin = 4/5 sin 53.1 =.7997 cos = 4/5 cos 36.9 =.7997 sin = cos 53.1 sin 53.1 = cos 36.9 3 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. 5 4 First, find the measure of LP using the sine function. Then, find the measure of LP using the cosine function. sine function cosine function 22 o P 12 Slide 154 / 240 L 68 o M Sine and osine are called co-functions of each other. o-functions of complementary angles are equal.
62 Given that sin 10 =.1736, write the cosine of a complementary angle. Slide 155 / 240 sin 10 =.1736 sin 80 =.9848 cos 10 =.9848 cos 80 =.1736 62 Given that sin 10 =.1736, write the cosine of a complementary angle. Slide 155 (nswer) / 240 sin 10 =.1736 sin 80 =.9848 nswer cos 10 =.9848 cos 80 =.1736 63 Given that cos 50 =.6428, write the sine of a complementary angle. Slide 156 / 240 sin 50 =.7660 sin 40 =.6428 cos 50 =.6428 cos 40 =.7660
63 Given that cos 50 =.6428, write the sine of a complementary angle. Slide 156 (nswer) / 240 sin 50 =.7660 sin 40 =.6428 cos 50 =.6428 cos 40 =.7660 nswer 64 Given that cos 65 =.4226, write the sine of a complementary angle. Slide 157 / 240 sin 25 =.4226 cos 25 =.9063 sin 65 =.9063 cos 65 =.4226 64 Given that cos 65 =.4226, write the sine of a complementary angle. Slide 157 (nswer) / 240 sin 25 =.4226 cos 25 =.9063 sin 65 =.9063 cos 65 =.4226 nswer
65 What can you conclude about the sine and cosine of 45 degrees? Students type their answers here Slide 158 / 240 65 What can you conclude about the sine and cosine of 45 degrees? Students type their answers here Slide 158 (nswer) / 240 nswer sin 45 = cos 45 Slide 159 / 240 Solving Right Triangles Return to the Table of ontents
To solve a right triangle means to find all 6 values in a triangle. Slide 160 / 240 The lengths of all 3 sides and the measures of all 3 angles. b c a 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. Slide 161 / 240 15 y 64 o z First, let's find the measure of. Slide 162 / 240 15 y 64 o z
First, let's find the measure of. Slide 162 (nswer) / 240 nswer 15 m< + m< = 90 o 64 o + m< = 90 64 o m< = 26 o z y Then, let's find the measure of. Slide 163 / 240 15 26 y 64 o z Then, let's find the measure of. Slide 163 (nswer) / 240 15 26 y nswer sin64 = y 15.8988 = y 15 y # 13.48 # 13.48 64 o z
Then, let's find the measure of. Slide 164 / 240 15 26 13.48 64 o z Then, let's find the measure of. Slide 164 (nswer) / 240 15 26 13.48 nswer a 2 + b 2 = c 2 64 o z 2 + (13.48) 2 = 15 2 z z 2 + 181.79 = 225 z 2 = 43.29 z 6.58 6.58 Try this... Find the missing parts of the triangle. Slide 165 / 240 R 11 E 37 o
Try this... Find the missing parts of the triangle. Slide 165 (nswer) / 240 R 11 E 37 o nswer R # 18.09 E # 14.36 m R = 53 o Let's solve a right triangle given the length of two sides (SS). Slide 166 / 240 9 15 y z First, find the length of since we know how to do that. ut, how do you find the measure of and? Slide 167 / 240 9 15 y z
First, find the length of since we know how to do that. ut, how do you find the measure of and? Slide 167 (nswer) / 240 nswer 9 a 2 + b 2 = c 2 z 2 + 9 2 = 15 2 z 2 + 81 = 225 z 2 = 15 144 z = 12 = 12 z y You will need to use the inverse trig functions. If sinθ =, θ = sin -1 If cosθ =, θ = cos -1 b c Pronounced inverse sine, inverse cosine, and inverse tangent. Slide 168 / 240 If tanθ =, θ = tan -1 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. a θ The 3 Inverse Trigonometric Ratios θ = sin -1 ( opposite side ) hypotenuse θ = cos -1 ( adjacent side ) hypotenuse θ = tan -1 ( opposite side ) adjacent side Slide 169 / 240 Use the inverse trig function to find the unknown angle measure when you know the length of 2 sides. Remember: b c a θ
66 Find sin -1 0.8. Round the angle measure to the nearest hundredth. Slide 170 / 240 66 Find sin -1 0.8. Round the angle measure to the nearest hundredth. Slide 170 (nswer) / 240 nswer θ = 53.13 67 Find tan -1 2.3. Round the angle measure to the nearest hundredth. Slide 171 / 240
67 Find tan -1 2.3. Round the angle measure to the nearest hundredth. Slide 171 (nswer) / 240 nswer θ = 66.50 68 Find cos -1 0.45. Round the angle measure to the nearest hundredth. Slide 172 / 240 68 Find cos -1 0.45. Round the angle measure to the nearest hundredth. Slide 172 (nswer) / 240 nswer θ = 63.26
To find an unknown angle measure in a right triangle, Slide 173 / 240 You need to identify the correct trig function that 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. θ 9 adj 15 hyp, I have a and h, so the ratio is a/h which is now you can solve for, the missing angle using the inverse trig function. How are you going to calculate the measure of? To find an unknown angle measure in a right triangle, Slide 173 (nswer) / 240 You need to identify the correct trig function that will find the missing value. Use SOHHTO to help. 9 9 is your angle of reference. 53.13 adj Label the two given sides θ of your triangle opp, adj, or hyp. 12 Identify the trig funtion that uses, 15 and the two sides. y 15 hyp Using, I have a and h, so the ratio is a/h which is cosine. nswer now you can solve for, the [This missing object a pull tab] angle using the inverse trig function. How are you going to calculate the measure of? Eample Find the trig equation that will find θ. Slide 174 / 240 7 θ 12
Eample Find the trig equation that will find θ. Slide 174 (nswer) / 240 θ 7 12 nswer adj 7 θ 12 opp Eample Find the trig equation that will find θ. Slide 175 / 240 θ 10 12 Eample Find the trig equation that will find θ. Slide 175 (nswer) / 240 θ 10 12 nswer adj θ 10 hyp 12
Eample Find the trig equation that will find θ. θ 12 Slide 176 / 240 9 Eample Find the trig equation that will find θ. Slide 176 (nswer) / 240 θ 9 12 nswer θ hyp 12 9 opp 69 Which is the correct trig equation to solve for Slide 177 / 240 7 12 E
69 Which is the correct trig equation to solve for Slide 177 (nswer) / 240 nswer 7 12 E 70 Which is the correct trig equation to solve for Slide 178 / 240 5 12 E 70 Which is the correct trig equation to solve for Slide 178 (nswer) / 240 nswer 5 12 E
71 Which is the correct trig equation to solve for Slide 179 / 240 K 11 J 9 L 71 Which is the correct trig equation to solve for Slide 179 (nswer) / 240 K nswer 11 J 9 L Try this... Solve the right triangle. Round your answers to the nearest hundredth. R Slide 180 / 240 24 7 Q S
Try this... Solve the right triangle. Round your answers to the nearest hundredth. R Slide 180 (nswer) / 240 24 7 Q S nswer QS = 25 m Q = 73.74 o m S = 16.26 o 72 Find E. Slide 181 / 240 5 E 8 72 Find E. Slide 181 (nswer) / 240 5 E Use the Pythagorean Theorem 8 nswer
73 Find m. Slide 182 / 240 5 E 8 73 Find m. Slide 182 (nswer) / 240 5 E Use inverse tangent 8 nswer 74 Find the m E. 5 E Slide 183 / 240 8
74 Find the m E. Slide 183 (nswer) / 240 5 E From before, 8 nswer 75 Find the m G. Slide 184 / 240 L 20 o 18 G 75 Find the m G. L Slide 184 (nswer) / 240 20 o nswer 18 G
76 Find L. L Slide 185 / 240 20 o 18 G 76 Find L. L Slide 185 (nswer) / 240 20 o nswer 18 G 77 Find the m P. 49.19 o P Slide 186 / 240 33.69 o 41.81 o 12 56.31 o E 18 N
77 Find the m P. 49.19 o P Slide 186 (nswer) / 240 33.69 o 41.81 o 56.31 o nswer E 18 12 N 78 Find RT. 10.44 S T Slide 187 / 240 12.45 11.47 9.53 8 40 o R 78 Find RT. 10.44 S T Slide 187 (nswer) / 240 12.45 11.47 9.53 nswer 8 40 o R
Slide 188 / 240 ngle of Elevation and epression Return to the Table of ontents How can you use trigonometric ratios to solve word problems involving angles of elevation and depression? Slide 189 / 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. Slide 190 / 240
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. The angle of elevation and the angle of depression are both measured relative to parallel horizontal lines, they are equal in meaure. Slide 192 / 240 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 none of the above
79 How can you describe the angle relationship between the angle of elevation and the angle of depression? Slide 193 (nswer) / 240 corresponding angles alternate interior angles alternate eterior angles none of the above nswer Eample Slide 194 / 240 my is flying a kite at an angle of 58 o. The kite's string is 158 feet long and my's arm is 3 feet off the ground. How high is the kite off the ground? 158 feet 58 o 3 feet sin# = 158 Slide 195 / 240 158ft 58 o sin58 =.8480 = = 134 158 158 Now, we must add in my's arm height. 134 + 3 = 137 The kite is about 137 feet off the ground.
Eample Slide 196 / 240 You are standing on a mountain that is 5306 feet high. You look down at your campsite at angle of 30 o. If you are 6 feet tall, how far is the base of the mountain from the campsite? 30 o 6 ft 5306 ft Slide 197 / 240 5312 ft tan30 =.5774 = 5312 5312 30 o.5774 = 5312 9,200 ft The campsite is about 9,200 ft from the base of the mountain. Try this... Slide 198 / 240 You are looking at the top of a tree. The angle of elevation is 55 o. 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?
Try this... Slide 198 (nswer) / 240 You are looking at the top of a tree. The angle of elevation is 55 o. 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? 55 o nswer 84 ft cos55 = 84 = 48.18 You are approimately 48 ft from the base [This object of the is a pull tree. tab] 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 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 (nswer) / 240 elevation depression nswer
81 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? Slide 200 / 240 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? 250 ft 345 ft Slide 200 (nswer) / 240 nswer ferry The angle of [This depression object is is a pull about tab] 46 degrees. 82 What is the distance from the ferry to the base of the statue? Slide 201 / 240
82 What is the distance from the crown ferry to the base of the statue? 250 ft 345 ft Slide 201 (nswer) / 240 nswer 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? Slide 203 / 240 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 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. Let's prove the Law of Sines 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 b h c a Statements with side lengths a, b, and c raw an altitude from to side Let h be the length of the altitude Reasons click Given ef of ltitude click click ef of sine Slide 205 / 240 Multiply click by b. Mult Prop of =. Multiply click by a. Mult Prop of =. Substitution click Prop of = ivide by ab. click ivision Prop of = Prove the Law of Sines (continued) Given: has sides of length a, b, and c c Prove: sin = sin = sin Statements a b c raw an altitude from to side b h Let g be the length of the altitude a b g c a Reasons click ef of ltitude Slide 206 / 240 click ef of sine Multiply click by c. Mult Prop of =. Multiply click by a. Mult Prop of =. Substitution click Prop of = ivide by ac. ivision Prop of = click Substitution click Prop of =
Use the Law of Sines to solve the triangle. a 70 o 10 sin = sin = sin a b c Select the ratios based on the given information. Given: m, m and (side c) (S) Slide 207 / 240 65 o b Which ratios must be used first? Why? Use the Law of Sines to solve the triangle. a 70 o 10 sin = sin = sin a b c Select the ratios based on the given information. Given: m, m and (side c) (S) Slide 207 (nswer) / 240 65 o b nswer Which ratios sin must = sinbe 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. a 70 o 10 Slide 208 / 240 65 o b sin = sin b c sin70 = sin65 b 10
First we can find the length side b. a 70 o 10 Slide 208 (nswer) / 240 65 o b sin = sin.9397 =.9063 b c b 10 sin70 = sin65 b 10.9063b = 9.397 nswer b 10.37 efore we find the length of side a, we find the m. a 70 o 10 Slide 209 / 240 65 o b=10.37 Triangle Sum Theorem m + m + m = 180 o efore we find the length of side a, we find the m. a 70 o 10 Slide 209 (nswer) / 240 65 o b=10.37 Triangle Sum Theorem m + m + m = 180 o nswer m + 70 o + 65 o = 180 o m + 135 o = 180 o m = 45 o
Now we find the length side a. Slide 210 / 240 a 70 o 10 65 o b=10.37 =45 o sin = sin a c Now we find the length side a. Slide 210 (nswer) / 240 a 70 o 10 sin = sin a c nswer 65 o =45 o sin45 b=10.37 = sin65 a 10.7071 =.9063 a 10.9063a = 7.071 a 7.8 Try this... Use the Law of Sines to find the length of side b (S). Slide 211 / 240 a b Since the length of the side opposite < is given, hint find the m< first. 85 o 29 o 9
Try this... Use the Law of Sines to find the length of side b (S). Slide 211 (nswer) / 240 a b Since the length of the side opposite < is given, hint find the m< first. 85 o 29 o 9 nswer b # 9.81 Eample... Find the length of side b (SS with an obtuse angle). Slide 212 / 240 b 8 101o 2.8 Eample... Find the length of side b (SS with an obtuse angle). sin = sin a c sin = sin101 8 2.8 8 sin =.9816 b 2.8 8 sin =.3436 =sin -1 (.3436) =20.09 m<+101+20.09=180 m<=58.91 sin = sin a 2.8b 101osin20.09 = sin58.91 2.8 b.3436 =.8564 2.8 b b = 6.98 Slide 212 (nswer) / 240
83 Find the m. Slide 213 / 240 19 o 70 o c 31 o 10 29 o 28 o 81 o b 83 Find the m. Slide 213 (nswer) / 240 19 o 70 o c 31 o 29 o 28 o 10 nswer 81 o b 84 Which ratio must be used to find the length of b or c? 70 o c 10 Slide 214 / 240 81 o b sin a sin b sin b sin c
84 Which ratio must be used to find the length of b or c? 70 o c 10 Slide 214 (nswer) / 240 81 o b sin a nswer sin b sin b sin c 85 What is the length of b? Slide 215 / 240 10 70 o c 81 o b 85 What is the length of b? Slide 215 (nswer) / 240 70 o 10 nswer c 81 o b
86 What is the length of c? Slide 216 / 240 10 70 o c 81 o b 86 What is the length of c? Slide 216 (nswer) / 240 c 70 o 10 81 o b nswer Law of osines c a Slide 217 / 240 If has sides of length a, b, and c, then: b 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).
Let's prove the Law of osines If has sides of length a, b, and c, then Given: has sides of length a, b, and c Prove: (similar reasoning shows that ) b c Statements with side lengths a, b, and c Given click raw an altitude from to side. Let h be the length of the alt. Let be the length of. Then (c-) is the length of. In, cos = /b (1) =b(cos) a b h Reasons click ef of ltitude a c- c Segment click ddition Postulate efinition click of cosine Multiply click by b. Mult Prop of =. (2) In, Pythagorean click Theorem In, Pythagorean click Theorem Slide 218 / 240 click Simplify click Substitution, equation (2) ssociative click Prop of ddition Substitution, click equation (1) Eample Use the Law of osines to solve the right triangle. Slide 219 / 240 a=16 c=27 a is opposite < b is opposite < c is opposite < b=23 The formula you choose depends on which angle you are solving for first. Slide 220 / 240 a=16 c=27 To find the m, b=23 a 2 = b 2 + c 2-2bc(cos) 16 2 = 23 2 + 27 2-2(23)(27)(cos) 256 = 529 + 729-1242(cos) 256 = 1258-1242(cos) -1002 = -1242(cos).8068 = cos = cos -1 (.8068) m 36.22 o
Slide 221 / 240 a=16 c=27 To find the m, 36.22 b=23 b 2 = a 2 + c 2-2ac(cos) 23 2 = 16 2 + 27 2-2(16)(27)(cos) 529 = 256 + 729-864(cos) 579 = 985-864(cos) -406 = -864(cos).4699 = cos =cos -1 (.4699) m 61.97 o or Using 2 different methods, the answers are slightly different because of rounding. Slide 222 / 240 61.97 a=16 c=27 36.22 b=23 To find the m, Use the Triangle Sum Theorem. Slide 222 (nswer) / 240 61.97 a=16 c=27 nswer 36.22 b=23 To find the m, m + m + m = 180 o 36.22 o + 61.97 o + m = 180 o Use the Triangle Sum Theorem. 98.19 o + m = 180 o m 81.81 o
Try this... Use the Law of osines to find the m< (SSS). Slide 223 / 240 7 6 5 Try this... Use the Law of osines to find the m< (SSS). Slide 223 (nswer) / 240 6 7 5 nswer b 2 = a 2 + c 2-2ac(cos) 7 2 = 5 2 + 6 2-2(5)(6)(cos) 49 = 25 + 36-60(cos) 49 = 61-60cos -12 = -60cos.2 = cos m< 78.46 o 87 In the triangle the length of c is... Slide 224 / 240 8 9 15 9 15 8
87 In the triangle the length of c is... Slide 224 (nswer) / 240 8 9 15 nswer 9 15 8 88 In the triangle the length of a is... Slide 225 / 240 8 9 15 9 15 8 88 In the triangle the length of a is... Slide 225 (nswer) / 240 8 9 15 nswer 9 15 8
89 Which formula would you use to find the m<? Slide 226 / 240 a 2 = b 2 + c 2-2ac(cos) a 2 = b 2 + c 2-2bc(cos) b 2 = a 2 + c 2-2ac(cos) c 2 = a 2 + b 2-2ab(cos) 89 Which formula would you use to find the m<? Slide 226 (nswer) / 240 a 2 = b 2 + c 2-2ac(cos) a 2 = b 2 + c 2-2bc(cos) nswer b 2 = a 2 + c 2-2ac(cos) c 2 = a 2 + b 2-2ab(cos) 90 What is the m? Slide 227 / 240 9 15 8
90 What is the m? Slide 227 (nswer) / 240 9 15 nswer 8 a 2 = b 2 + c 2-2bc(cos) 8 2 = 9 2 + 15 2-2(9)(15)(cos) 64 = 81 + 225-270(cos) 64 = 306-270cos -242 = -270cos.8963 = cos m 26.32 o 91 What is the m? Slide 228 / 240 9 15 8 91 What is the m? Slide 228 (nswer) / 240 9 15 nswer 8 c 2 = a 2 + b 2-2ab(cos) 15 2 = 8 2 + 9 2-2(8)(9)(cos) 225 = 64 + 81-144(cos) 225 = 145-144cos 80 = -144cos -.5556 = cos =cos -1 (-.5556) m 123.75 o
92 What is the measure of (S)? Students type their answers here 4 Slide 229 / 240 50 8 92 What is the measure of (S)? Students type their answers here a 2 = b 2 + c 2-2bc(cos50) a 4 2 = 8 2 + 4 2-2(8)(4)(.6428) a 2 = 64 + 16-41.1384 50 a 2 = 38.8616 a = 6.23 8 nswer or b 2 = a 2 + c 2-2ac(cos) 8 2 = 6.23 2 + 4 2-2(6.23)(4)(cos) 64 = 38.8129 + 16-49.84(cos) 49 = 54.8129-49.84cos -5.8129 = -49.84cos.1166 = cos m 83.3 o Slide 229 (nswer) / 240 93 The Law of Sines and osines is used to solve... Slide 230 / 240 right triangles acute triangles obtuse triangles all triangles
93 The Law of Sines and osines is used to solve... Slide 230 (nswer) / 240 right triangles acute triangles obtuse triangles all triangles nswer Slide 231 / 240 rea of an Oblique Triangle Return to the Table of ontents o you remember this? Previously, we found the area of a triangle when we were given 3 sides. Slide 232 / 240 Find the area of the triangle. 13 feet 13 feet 10 feet
1 = 2 bh 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 - Slide 233 / 240 13 feet h 13 feet 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 67.38 10 feet Since = 1 bh and b = 10, we need to find h. 2 Slide 235 / 240 13 feet h 67.38 10 feet
Let's derive the formula for an oblique triangle. Given: has sides of length a, b, and c. ltitude h. Prove: Slide 236 / 240 a h b c Statements Reasons with side lengths a, b, and c Given click raw an altitude from to side ef clickof ltitude Let h be the length of the altitude ef clickof sine Multiply click by a. Mult Prop of =. efinition. Formula for the click area of a triangle. Substitution click Prop of = ommutative Prop of Multiplication click Slide 237 / 240 Slide 237 (nswer) / 240
95 Find the area of the triangle to the nearest tenth. Students type their answers here Slide 238 / 240 95 Find the area of the triangle to the nearest tenth. Students type their answers here Slide 238 (nswer) / 240 nswer 96 Find the area of the triangle to the nearest tenth. Students type their answers here Slide 239 / 240
96 Find the area of the triangle to the nearest tenth. Students type their answers here Slide 239 (nswer) / 240 nswer 97 Find the area of the triangle to the nearest tenth. Students type their answers here Slide 240 / 240 97 Find the area of the triangle to the nearest tenth. Students type their answers here Slide 240 (nswer) / 240 or nswer