Slide 1 / 240 New Jersey Center for Teaching and Learning 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. NJCTL 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. Click to go to website: www.njctl.org
Slide 2 / 240 Geometry Trigonometry of Right Triangles 2014-06-05 www.njctl.org
Slide 3 / 240 Table of Contents Pythagorean Theorem Similarity in Right Triangles Special Right Triangles Trigonometric Ratios Solving Right Triangles Angles of Elevation and Depression Law of Sines and Law of Cosines Area of an Oblique Triangle Click on a Topic to go to that section
Slide 4 / 240 Pythagorean Theorem Return to the Table of Contents
Slide 5 / 240 Before learning about similar right triangles and trigonometry, we need to review the Pythagorean Theorem and the Pythagorean Theorem Converse.
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.
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 Example: x 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 9 / 240 Solve for x: 92 + 122 = x2 x 9 81 + 144 = x2 225 = x2 15 = x -15 is a extraneous solution, a distance can not equal a negative number. x = 15 12
Slide 10 / 240 x 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 Example:
Slide 11 / 240 1 The missing side is the of the right triangle. B hypotenuse 6 9 x A leg
Slide 12 / 240 2 Find the length of the missing side. 6 9 x
Slide 13 / 240 A leg B hypotenuse x 36 15 3 The missing side is the of the right triangle.
Slide 14 / 240 x 36 15 4 Find the length of the missing side.
Slide 15 / 240 Real World Application 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? 28 feet 7 feet The ladder will reach feet up the wall safely. 2 Solve using a + b = c
Slide 17 / 240 Real World Application 50 x 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
Slide 18 / 240 5 A NBA 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? A 94.03 feet B 117.7 feet C 118 feet D 94 feet (Round the answer to the nearest whole number)
Slide 19 / 240 Pythagorean Theorem Applications The Pythagorean Theorem can also be used in figures that contain right angles.
Slide 20 / 240 Example Find the perimeter of the square. 18 cm Psq = 4s note: Before 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. 18 cm x Start here: 2 x 2 + x 2 = 18
Slide 22 / 240 Example Find the area of the triangle. The base of the triangle is given, but we need to find the height of the triangle. A = 1 2 bh 13 feet 13 feet 10 feet
Slide 23 / 240 13 feet h 5 feet 13 feet 5 feet By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base.
Slide 24 / 240 Try this... Find the perimeter of the rectangle. 10 in P rect = 2l + 2w 8 in
Slide 25 / 240 6 Find the area of the rectangle. B 84 square feet C 46 square inches D 46 square feet 8 feet 1 ee 7f t A 120 square feet
Slide 26 / 240 7 Find the perimeter of the square. (Round to the nearest tenth) D 36 cm cm C 25.6 cm 9 B 25.5 cm A 12.8 cm
Slide 27 / 240 7 inches 7 inches 10 inches 8 Find the area of the triangle.
Slide 28 / 240 9 Find the area of the triangle. 7 inches 4 inches 7 inches
Slide 29 / 240 Converse 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. B If c = a + b, then ABC is a right triangle. 2 2 2 a C c b A
Slide 30 / 240 Example Tell whether the triangle is a right triangle. Remember c is the longest side E 7 25 F D 24
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. B If c > a + b, then ABC is obtuse. 2 2 c a 2 C b A
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. B a c If c2 < a2 + b2, then ABC is acute. A b C
Slide 33 / 240 Example Classify the triangle as acute, right, or obtuse. 15 17 13
Slide 34 / 240 10 Classify the triangle as acute, right, obtuse, or not a triangle. 12 B right C obtuse D not a triangle 15 11 A acute
Slide 35 / 240 A acute B right C obtuse D not a triangle 5 3 6 11 Classify the triangle as acute, right, obtuse, or not a triangle.
Slide 36 / 240 A acute 25 B right C obtuse D not a triangle 20 19 12 Classify the triangle as acute, right, obtuse, or not a triangle.
Slide 37 / 240 13 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. B right C obtuse A acute
Slide 38 / 240 A acute triangle B right triangle C obtuse triangle 14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle.
Slide 39 / 240 Review If c = a + b, then triangle is right. 2 2 2 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 Contents
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. C A B D CD is the altitude of ABC~ ACD~ ABC CBD
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
Slide 44 / 240 C Let's prove the Theorem. 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: ABC is a right triangle is the altitude of ABC ABC~ ACD~ CBD A B D Statements ABC is a right triangle is a right angle Reasons Given click Given click Def clickof Altitude is a right angle Def of Perp Lines. 2 lines that form a rtclick angle All rt angles are click Reflexive Prop of click ABC ~ ACD is a right angle AA~ click Def of Perp Lines click All rt angles are click Reflexive Prop of click ABC ~ CBD ABC~ ACD~ CBD AA~ click Transitive Prop of ~ click
Slide 45 / 240 C Let's sketch the 3 triangle's separately, with the same orientation. A B B C C A A D C B 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. 30 B B C 30 30 60 30 60 A 60 60 60 C A 30 D C D
Slide 46 / 240 e c a Assign lengths to all the segments. Let the lengths of the segments on the hypotenuse be d and e. B B c d b Label the sides of a triangle with the lower case letter of the opposite angle. A b C a C a b A d D C e ABC~ D ACD~ Because the triangles are similar the corresponding sides are proportional. ABC~ ACD ABC~ CBD CBD
Slide 47 / 240 To prove the Pythagorean Theorem, use the proportions. Given: ABC~ ACD click Using the multiplication property of equality, multiply the equation by bc. click Prove: (1) e a Altitude of a rt triangle click theorem. Definition of similar triangles. ABC is a right triangle. is an altitude. c Reasons Statements ABC~ CBD simplify click Altitude of a rt triangle click theorem. Definition of similar triangles. click d Using the multiplication property of equality, click multiply the equation by ac b (2) simplify click
Slide 48 / 240 To prove the Pythagorean Theorem, use the proportions (continued). Given: ABC is a right triangle. is an altitude. Prove: Statements 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 49 / 240 Example Find the length of the altitude KI? H 13 12 K I 5 J
Slide 50 / 240 H It maybe helpful to sketch the 3 triangle's separately, with the same orientation. H H 13 12 I x K K 5 5 J J 12 x I K x 13 I J 12 5 K Because the triangles are similar the corresponding sides are proportional. 13x = 60 x 4.62
Slide 51 / 240 Try this... Find the length of RS. 3 P Q R 4 5 x S 5 P x 4 3 R R Q S 4 Q P 5 S S
Slide 52 / 240 15 Which ratio is the ratio of corresponding sides? A B C D H K J I
Slide 53 / 240 16 Find KJ. I 7 J K 25 H 24
Slide 54 / 240 The next two theorems are Geometric Mean Theorems. What is a mean? An 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: Arithmetic Mean vs Geometric Mean
Slide 55 / 240 The geometric mean of two positive numbers a and b is the positive number x that satisfies a =x x b x2 = ab x= 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 Example Find the geometric mean of 8 and 14. x2 = 8(14) x2 = 112 (only the positive value)
Slide 57 / 240 17 Find the geometric mean of 7 and 56. Write the answer is simplest radical form. B C D A
Slide 58 / 240 18 Find the geometric mean of 3 and 48. Students type their answers here
Slide 59 / 240 Corollary 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. C CD is the altitude of Since, A D ACD~ CBD B CD2 = AD(DB) ABC
Slide 60 / 240 Example Find z. 8 6 z
Slide 61 / 240 Example Find z. 18 6 z
Slide 62 / 240 Try this... Find y. 8 18 y 2) 12 9 1) y
Slide 63 / 240 19 Find x. 10 C 20 D 50 5 B 100 x A
Slide 64 / 240 20 Find x. A 99 C D 9 x B 11
Slide 65 / 240 Corollary 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. C A D CD is the altitude of B Since, ABC~ ABC~ ACD AD AC = AC AB ABC ACD~ CBD ABC~ CBD AB BC = BC DB
Slide 66 / 240 Example 4 R S 9 T Find x. x U
Slide 67 / 240 Example D E 4 F 6 G Find x. x
Slide 68 / 240 21 Is PR a geometric mean between QR and SR? True False Q S R P
Slide 69 / 240 22 Is the geometric mean correct? P False Q S R True
Slide 70 / 240 23 Which proportion is correct? K L M A B C D J
Slide 71 / 240 24 Find y. B 20 C 5 D 12 y 9 16 A
Slide 72 / 240 25 Find y. B 18 C 24 D None of the above 27 y x 9 A 3
Slide 73 / 240 26 Find x. 5 8 x
Slide 74 / 240 Special Right Triangles Return to the Table of Contents
Slide 75 / 240 In this section you will learn about the properties of the two special right triangles. 45-45-90 30-60-90 45o 90o 60o 45o 90o 30o
Slide 76 / 240 45-45-90 Triangle Theorem hypotenuse = leg( 2) 45o Can you prove this? x 2 x x 45o A 45-45-90 triangle is an isosceles right triangle, where the hypotenuse is 2 times the length of the leg.
Slide 77 / 240 P 6 Q 45o y x 45o R Example Find the length of the missing sides. Write the answer in simplest radical form.
Slide 78 / 240 y S T x 18 V Example Find the length of the missing sides of the right triangle.
Slide 79 / 240 Try this... Find the length of the missing sides. y 8 x
Slide 80 / 240 27 Find the value of x. A 5 C (5 2)/2 y 5 B 5 2 x
Slide 81 / 240 28 Find the value of y. A 5 B 5 2 C (5 2)/2 y 5 x
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 83 / 240 30 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm.
Slide 84 / 240 30-60-90 Triangle Theorem 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. x 60o hypotenuse = 2(shorter leg) longer leg = 3(shorter leg) 2x 30o x 3
Slide 85 / 240 This can be proved using an equilateral triangle. x For right triangle ABD, BD is a perpendicular bisector. let a = x, c = 2x and b= BD B c=2x A 60 a=x 2x x 3 30 30 2x b D 60o x 60 C 30o
Slide 86 / 240 G 30o Example Find the length of the missing sides of the right triangle. x y H 60o 5 F
Slide 87 / 240 G x HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF 30o Recall triangle inequality, the shortest side is opposite the smallest angle and the longest side is opposite the largest angle. y H 60o 5 F
Slide 88 / 240 x M 60o y 9 30o T A Example Find the length of the missing sides of the right triangle.
Slide 89 / 240 Example Find the area of the triangle. 14 ft
Slide 90 / 240 The altitude (or height) divides the triangle into two 30o-60o-90o triangles. h? The length of the shorter leg is 7 ft. The length of the longer leg is 7 3 ft. A = b(h) = 14(7 3) A 84.87 square ft 14 ft?
Slide 91 / 240 Try this... Find the length of the missing sides of the right triangle. 15 30o y 60o x
Slide 92 / 240 9 ft 30o Try this... Find the area of the triangle.
Slide 93 / 240 31 Find the value of x. B 7 3 C (7 2)/2 D 14 60o 7 30o x A 7
Slide 94 / 240 32 Find the value of x. B 7 3 C (7 2)/2 D 14 x 7 2 A 7
Slide 95 / 240 33 Find the value of x. B 7 3 C (7 2)/2 D 14 30o x 60o A 7 7 3
Slide 96 / 240 34 The hypotenuse of a 30o -60o -90o triangle is 13 cm. What is the length of the shorter leg?
Slide 97 / 240 35 The length the longer leg of a 30o -60o -90o triangle is 7 cm. What is the length of the hypotenuse?
Slide 98 / 240 Real World Example 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? 30o 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. 2.5
Slide 100 / 240 45o? 3 feet 36 A 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 101 / 240 45o 3 feet? 37 What is the length of the base of the ramp? Round to the nearest hundredth.
Slide 102 / 240 38 The yield sign is shaped like an equilateral triangle. Find the length of the altitude. 20 inches
Slide 103 / 240 39 The yield sign is shaped like an equilateral triangle. Find the area of the sign. 20 inches
Slide 104 / 240 Trigonometric Ratios Return to the Table of Contents
Slide 105 / 240 Right triangle trigonometry is the study of the relationships between the sides and angles of right triangles. A c b C a B
Slide 106 / 240 Ever since the construction of the Bell 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, Bell Tower in Pisa, Italy
Slide 107 / 240 Engineers can measure the angle of slant using any of the right triangles constructed below. ABC~ DBE~ FBG A D F WHY? BGEC angle of slant Engineers very carefully measure the perpendicular distance from a tower window (points A, D or F) to the ground (points G, E or C). Then they measure the distance from the tower to points C, E or G.
Slide 108 / 240 Triangle Height Base Ratio Height / Base ABC AC=50m BC=5m 50/5=10 DBE DE=30m BE=3m 30/3=10 FBG FG=20m BG=2m 20/2=10 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. Let's calculate the ratio's of the height to the base for each right triangle.
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? Click 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, A and B are the acute angles. A Label the sides of a triangle with the lower case letter of the opposite angle. c b C a B
Slide 111 / 240 A Let's look at A, when A is the reference angle, the side opposite A is a. the side adjacent (or next to) A is b. b and the hypotenuse is c. adj A C c b opp C adj a hyp B c hyp a opp When B is the reference angle, the side opposite B is b. the side adjacent (or next to) B is a. and the hypotenuse is c. B
Slide 112 / 240 40 What is the side opposite to J? B LK J L C KJ K A JL
Slide 113 / 240 41 What is the hypotenuse of the triangle? J L B LK C KJ K A JL
Slide 114 / 240 42 What is the side adjacent to J? A JL J L C KJ K B LK
Slide 115 / 240 43 What is the side opposite K? J L B LK C KJ K A JL
Slide 116 / 240 44 What is the side adjacent to K? J L B LK C KJ K A JL
Slide 117 / 240 Trigonometric Ratios A 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). A c b C a B
Slide 118 / 240 The 3 Trigonometric Ratios sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse opposite side adjacent side tanθ = A This spells... SOHCAHTOA or which is a pneumonic to help you remember the sides of a right triangle (you'll need to remember the spelling). θ c b C a B
Slide 119 / 240 Click for a SOHCAHTOA song on youtube.com "Gettin' Triggy Wit It".
Slide 120 / 240 D Example Find the sin F, cos F, and tan F. 10 6 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. Always reduce D opp 10 6 E adj 8 opp 6 fractions to lowest terms. 3 sinf = hyp = 10 = 5 hyp F cosf = adj = 8 = 4 tan F hyp 10 = opp = 6 8 adj 5 = 3 4
Slide 121 / 240 D Example Find the sin D, cos D, and tan D. 10 6 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. Always reduce D adj 10 6 E opp 8 opp 8 fractions to lowest terms. 4 sind = hyp = 10 = 5 hyp F cosd = adj = 6 = 3 hyp tand = 10 5 opp = 8 = 4 3 6 adj
Slide 122 / 240 45 What is the sin R? B 21/20 C 21/29 D 20/21 A 20/29
Slide 123 / 240 46 What is the cosr? B 21/20 C 21/29 D 20/21 A 20/29
Slide 124 / 240 47 What is the tanr? B 21/20 C 20/29 D 21/29 A 20/21
Slide 125 / 240 48 What is the sinq? B 21/20 C 21/29 D 29/20 A 20/29
Slide 126 / 240 A 20/29 B 21/20 C 21/29 D 29/21 49 What is the cosq?
Slide 127 / 240 50 What is the tanq? B 21/20 C 21/29 D 20/21 A 20/29
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. Check that your calculator is set for degrees (not radians) and round your answer to the ten thousandth place (4 decimal places). Find the following: click sin 84.3 =.9951 cos 84.3 =.0993 click tan 84.3 = 10.0187 click A D F B angle of slant C
Slide 129 / 240 A 0.5 B 0.8660 C 1.7321 D 0.5774 51 Evaluate sin 60. Round to the nearest ten thousandth.
Slide 130 / 240 52 Evaluate cos 60. Round to the nearest ten thousandth. B 0.8660 C 1.7321 D 0.5774 A 0.5
Slide 131 / 240 53 Evaluate tan 60. Round to the nearest ten thousandth. B 0.8660 C 1.7321 D 0.5774 A 0.5
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. x
Slide 133 / 240 How do you find an unknown side measure in a right triangle when you are given an acute angle and one side? You need to identify the correct trig function that will find the missing side. Use SOHCAHTOA to help. B is your angle of reference. Label the given and unknown sides of your triangle opp, adj, or hyp. Identify the trig funtion that uses B, 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 x, the missing side. opp adj
Slide 134 / 240 Example Find the trig equation that will find x. x C 30o 12 A B
Slide 135 / 240 Example Find the trig equation that will find x. B C 12 A x 30o
Slide 136 / 240 Example Find the trig equation that will find x. 30o C 12 x A B
Slide 137 / 240 54 Using B, which is the correct trig equation needed to solve for x. A sin40 = 12/x o 0 4 B cos40 = x/12 C tan40 = 12/x x 12 B D sin40 = x/12 E D
Slide 138 / 240 55 Using D, which is the correct trig equation needed to solve for x. B B cos50 = x/12 C tan50 = 12/x x D sin50 = x/12 E A sin50 = 12/x 12 50o D
Slide 139 / 240 A tan32 = x/11 11 B cos32 = x/11 C tan32 = 11/x D sin32 = 11/x K J 32o x L 56 Using J, which is the correct trig equation needed to solve for x.
Slide 140 / 240 58 o A tan58 = x/11 B cos58 = x/11 C tan58 = 11/x D sin 58 = 11/x J x K 11 L 57 Using K, which is the correct trig equation needed to solve for x.
Slide 141 / 240 Finding the Missing Side of a Right Triangle Now, you can solve for x, 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 next slide). adj
Slide 142 / 240 Finding the Missing Side of a Right Triangle Now, you can solve for x, 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 Example Find x. Round your answer to the nearest hundredth. 25 o E x 12 M G
Slide 144 / 240 Example Find x. Round your answer to the nearest hundredth. x 12 o 65 M E G
Slide 145 / 240 Example Find y. Round your answer to the nearest hundredth. C 10 20o E y A
Slide 146 / 240 P 12 L 68o M 58 Find the length of LM. Round your answer to the nearest tenth.
Slide 147 / 240 P 12 L 68o M 59 Find the length of LP. Round your answer to the nearest tenth.
Slide 148 / 240 Explain and use the relationship between the sine and cosine of complementary angles.
Slide 149 / 240 Find the measure of A?
Slide 150 / 240 To find the measure of A... The sum of the interior angles of any triangle is equal to 180 degrees. A and B are complementary angles. Complementary angles are two angles whose sum of their measures is 90 degrees. The acute angles of a right triangle are always complementary.
Slide 151 / 240 60 For right triangle ABC, what is the measure of B? B A 30 degrees B 50 degrees C 60 degrees D cannot be determined A 30o C
Slide 152 / 240 61 If the, find the complementary angle? B 70 degrees C 160 degrees D none of the above A 20 degrees
Slide 153 / 240 B 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 A + m B = 90 36.9 53.1 + 36.9 = 90 5 4 sin A = 4/5 sin 53.1 =.7997 cos B = 4/5 cos 36.9 =.7997 sin A = cos B 53.1 sin 53.1 = cos 36.9 C A 3 The sine of an angle is equal to the cosine of its complement. cos A = 3/5 cos 53.1 =.6004 sin B = 3/5 sin 36.9 =.6004 cos A = sin B cos 53.1 = sin 36.9 The cosine of an angle is equal to the sine of its complement.
Slide 154 / 240 First, find the measure of LP using the sine function. Then, find the measure of LP using the cosine function. sine function cosine function x L 68o Sine and Cosine are called co-functions of each other. Co-functions of complementary angles are equal. P 22o 12 M
Slide 155 / 240 62 Given that sin 10 =.1736, write the cosine of A sin 10 =.1736 B sin 80 =.9848 C cos 10 =.9848 D cos 80 =.1736 a complementary angle.
Slide 156 / 240 63 Given that cos 50 =.6428, write the sine of A sin 50 =.7660 B sin 40 =.6428 C cos 50 =.6428 D cos 40 =.7660 a complementary angle.
Slide 157 / 240 64 Given that cos 65 =.4226, write the sine of a complementary angle. B cos 25 =.9063 C sin 65 =.9063 D cos 65 =.4226 A sin 25 =.4226
Slide 158 / 240 65 What can you conclude about the sine and cosine of 45 degrees? Students type their answers here
Slide 159 / 240 Solving Right Triangles Return to the Table of Contents
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. A c b C a B
Slide 161 / 240 Let's solve a right triangle given the length of one side and the measure of one acute an gle (AAS). You need to find the 3 missing parts. A x 15 C 64o y z B
Slide 162 / 240 First, let's find the measure of A. A x 15 z B C 64o y
Slide 163 / 240 Then, let's find the measure of AB. A 26 15 z B C 64o y
Slide 164 / 240 Then, let's find the measure of BC. A 26 15 z B C 64o 13.48
Slide 165 / 240 Try this... Find the missing parts of the triangle. 11 E 37o D R
Slide 166 / 240 Let's solve a right triangle given the length of two sides (SSA). A 9 B x z 15 y C
Slide 167 / 240 First, find the length of BC since we know how to do that. But, how do you find the measure of A and C? 9 B x z 15 y C A
Slide 168 / 240 You will need to use the inverse trig functions. If sinθ =, θ = sin If cosθ =, θ = cos-1 If tanθ =, θ = tan-1 A -1 c b C Pronounced inverse sine, inverse cosine, and inverse tangent. a θ B 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.
Slide 169 / 240 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. A Remember: c b C a θ B
Slide 170 / 240 66 Find sin-1 0.8. 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 172 / 240 68 Find cos-1 0.45. Round the angle measure to the nearest hundredth.
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 SOHCAHTOA to help. Using cosine. 9 adj θ 15 hyp, I have a and h, so the ratio is a/h which is now you can solve for A, the missing angle using the inverse trig function. How are you going to calculate the measure of C? B A is your angle of reference. Label the two given sides of your triangle opp, adj, or hyp. Identify the trig funtion that uses A, and the two sides. A C
Slide 174 / 240 Example Find the trig equation that will find θ. B 7 C 12 A θ
Slide 175 / 240 Example Find the trig equation that will find θ. 10 C θ 12 A B
Slide 176 / 240 Example Find the trig equation that will find θ. θ C 12 9 A B
Slide 177 / 240 69 Which is the correct trig equation to solve for B A 7 12 B C D E D
Slide 178 / 240 70 Which is the correct trig equation to solve for B B 5 C D E 12 A D
Slide 179 / 240 A K B 11 C D J 9 L 71 Which is the correct trig equation to solve for
Slide 180 / 240 Try this... Solve the right triangle. Round your answers to the nearest hundredth. R Q 7 S 24
Slide 181 / 240 72 Find CE. E D 5 8 C
Slide 182 / 240 73 Find m C. E D 5 8 C
Slide 183 / 240 74 Find the m E. E D 5 8 C
Slide 184 / 240 75 Find the m G. o 20 L A 18 G
Slide 185 / 240 o 20 76 Find AL. L A 18 G
Slide 186 / 240 77 Find the m P. P A 49.19o 12 C 41.81o D 56.31o E 18 N B 33.69o
Slide 187 / 240 78 Find RT. S T B 12.45 C 11.47 D 9.53 8 40o R A 10.44
Slide 188 / 240 Angle of Elevation and Depression Return to the Table of Contents
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.
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.
Slide 193 / 240 79 How can you describe the angle relationship between the angle of elevation and the angle of depression? C alternate exterior angles D none of the above A corresponding angles B alternate interior angles
Slide 194 / 240 Example Amy is flying a kite at an angle of 58o. The kite's string is 158 feet long and Amy's arm is 3 feet off the ground. How high is the kite off the ground? 15 x 8f ee t 58o 3 feet
Slide 195 / 240 x 15 8f t 58o sinθ = x 158 sin58 = x 158.8480 = x 158 x = 134 Now, we must add in Amy's arm height. 134 + 3 = 137 The kite is about 137 feet off the ground.
Slide 196 / 240 Example 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? 30o 6 ft 5306 ft x
Slide 197 / 240 tan30 = 5312 ft 30o x.5774 = 5312 x 5312 x.5774x = 5312 x 9,200 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?
Slide 199 / 240 A elevation B 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 200 / 240 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 201 / 240 82 What is the distance from the ferry to the base of the statue?
Slide 202 / 240 Law of Sines and Law of Cosines Return to the Table of Contents
Slide 203 / 240 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 Cosines 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 (AAS or ASA) 2. Two side lengths and the measure of a non-included angle (SSA) when the angle is a right angle or an obtuse angle. The Law of Sines has a problem dealing with SSA when the angle is acute. There can be zero, one or two solutions. Click on: Khan Academy Video "More On Why SSA Is Not A Postulate" for more info. You can use the Law of Cosines when you are given 3. Three side lengths (SSS) 4. Two side lengths and the measure of an included angle (SAS)
Slide 204 / 240 C Law of Sines b A a c If ABC has sides of length a, b, and c, then sin A = sinb = sinc a b c To use the Law of Sines, 2 angles and 1 side must be given. B
Slide 205 / 240 C Let's prove the Law of Sines If ABC has sides of length a, b, and c, then sin A = sinb = sinc A a b c Given: ABC has sides of length a, b, and c Prove: sin A = sinb = sinc a b c b h c a B Statements Reasons ABC with side lengths a, b, and c click Given Def of Altitude Draw an altitude from C to side AB 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
Slide 206 / 240 C Prove the Law of Sines (continued) Given: ABC has sides of length a, b, and c Prove: sin A = sinb = sinc a b c b A h c C a b g B A Statements a c B Reasons Def of Altitude Draw an altitude from B to side AC 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
Slide 207 / 240 Use the Law of Sines to solve the triangle. sin A = sinb = sinc a b c Select the ratios based on the given information. B C 70o 65o b Given: m B, m C and BA (side c) (AAS) 10 A Which ratios must be used first? Why? a
Slide 208 / 240 C sinb = sinc b c sin70 = sin65 b 10 a 70o 65o b 10 A First we can find the length side b. B
Slide 209 / 240 C B a 70o 65o Triangle Sum Theorem m A + m B + m C = 180o b=10.37 10 A Before we find the length of side a, we find the m A.
Slide 210 / 240 B a C sina = sinc a c 65o 70o 10 =45o b=10.37 A Now we find the length side a.
Slide 211 / 240 Try this... Use the Law of Sines to find the length of side b (ASA). C 85o 9 29o A b a B Since the length of the side opposite <C is given, find the m<c first. hint
Slide 212 / 240 Example... Find the length of side b (SSA with an obtuse angle). 8 A 101o b 2.8 C B
Slide 213 / 240 83 Find the m A. B 31o C 29o D 28o 70o c A 10 81o C b A 19o B
Slide 214 / 240 84 Which ratio must be used to find the length of B b or c? 70o c 81o A sina a B sina b C b C sinb b D sinc c A 10
Slide 215 / 240 85 What is the length of b? B c A 10 81o C b 70o
Slide 216 / 240 86 What is the length of c? B c A 10 81o C b 70o
Slide 217 / 240 B Law of Cosines c A a b C If ABC has sides of length a, b, and c, then: To use the Law Cosines, you must be given the length of 3 sides (SSS) or the length of 2 sides and the measure of the included angle (SAS).
Slide 218 / 240 Let's prove the Law of Cosines A Given: ABC has sides of length a, b, and c Prove: (similar reasoning shows that ) b a b If ABC has sides of length a, b, and c, then C C c B A Statements h x a D c-x c B Reasons ABC with side lengths a, b, and c Given click Def of Altitude Draw an altitude CD from C to side click AB. Let h be the length of the alt. Let x be the length of AD. Then (c-x) is the length of DB. Segment Addition Postulate click In Definition of cosine click ADC, cosa = x/b (1) x=b(cosa) Multiply click by b. Mult Prop of =. (2) In Pythagorean Theorem click In ADC, CDB, Pythagorean Theorem click Simplify click Substitution, equation (2) click Associative Prop of Addition click Substitution, equation (1) click
Slide 219 / 240 Example Use the Law of Cosines to solve the right triangle. B a=16 C a is opposite <A b is opposite <B c is opposite <C c=27 b=23 A The formula you choose depends on which angle you are solving for first.
Slide 220 / 240 B c=27 a=16 To find the m A, C b=23 a2 = b2 + c2-2bc(cosa) 162 = 232 + 272-2(23)(27)(cosA) 256 = 529 + 729-1242(cosA) 256 = 1258-1242(cosA) -1002 = -1242(cosA).8068 = cosa A = cos-1(.8068) m A 36.22o A
Slide 221 / 240 B a=16 c=27 36.22 To find the m B, C b2 = a2 + c2-2ac(cosb) 232 = 162 + 272-2(16)(27)(cosB) 529 = 256 + 729-864(cosB) 579 = 985-864(cosB) -406 = -864(cosB).4699 = cosb B=cos-1(.4699) m B 61.97o b=23 A or Using 2 different methods, the answers are slightly different because of rounding.
Slide 222 / 240 B C c=27 61.97 36.22 b=23 To find the m C, Use the Triangle Sum Theorem. A a=16
Slide 223 / 240 Try this... Use the Law of Cosines to find the m<b (SSS). 7 A 6 5 B C
Slide 224 / 240 87 In the triangle the length of c is... B 9 C 15 A 15 9 C 8 B A 8
Slide 225 / 240 88 In the triangle the length of a is... B 9 C 15 A 15 9 C 8 B A 8
Slide 226 / 240 89 Which formula would you use to find the m<a? B a2 = b2 + c2-2bc(cosa) C b2 = a2 + c2-2ac(cosb) D c2 = a2 + b2-2ab(cosc) A a2 = b2 + c2-2ac(cosa)
Slide 227 / 240 90 What is the m A? A 15 C 8 B 9
Slide 228 / 240 91 What is the m C? A 9 C 8 B 15
Slide 229 / 240 92 What is the measure of B (ASA)? Students type their answers here B 4 50 8 C A
Slide 230 / 240 93 The Law of Sines and Cosines is used to solve... B acute triangles C obtuse triangles D all triangles A right triangles
Slide 231 / 240 Area of an Oblique Triangle Return to the Table of Contents
Slide 232 / 240 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 233 / 240 A = 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 - 13 feet h 5 feet 13 feet 5 feet
Slide 234 / 240 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 (SAS)? Find the area of the triangle. 13 feet 67.38 10 feet
Slide 235 / 240 Since A = 1 bh 2 and b = 10, we need to find h. 13 feet h 67.38 10 feet
Slide 236 / 240 Let's derive the formula for an oblique triangle. Given: ABC has sides of lengtha, b, and c. Altitude h. Prove: B a C h Statements Reasons click ABC with side lengths a, b, and c Given Draw an altitude from B to side AC Def clickof Altitude c Let h be the length of the altitude b A Def clickof sine Multiply click by a. Mult Prop of =. Definition. Formula for the area of a triangle. click Substitution Prop of = click Commutative Prop of Multiplication click
Slide 237 / 240 A D B E C F 94 Which of the following expressions can be used to find the area of the triangle below? Select all that apply.
Slide 238 / 240 95 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 240 / 240 97 Find the area of the triangle to the nearest tenth. Students type their answers here