UNIT 9 - RIGHT TRIANGLES AND TRIG FUNCTIONS Converse of the Pythagorean Theorem Objectives: SWBAT use the converse of the Pythagorean Theorem to solve problems. SWBAT use side lengths to classify triangles by their angle measures. Converse of the Pythagorean Theorem If a b c works, then the triangle is a right triangle. Examples: Determine whether the triangle is a right triangle. 10 1... 5 8 7 1 1 5 10 1 5 100 144 15 144 No Along the same lines 8 7 1 64 49 169 11 169 No 8 4 4 4 4 8 64 64 64 Yes Acute Theorem~ Obtuse Theorem~ If c < b + a, then it is an acute triangle If c > b + a, then it is an obtuse triangle Adding b + a is larger than c Adding b + a is smaller than c
Examples: Classify the triangle as acute, obtuse, or right (if possible). 4) 5, 1, 1 5) 7,14,16 5 1 1 5 144 169 169 169 Right Triangle 7 14 16 49 196 56 45 56 c is larger Obtuse Triangle 6) 6, 1, 0 7) 10, 11, 1 6 1 0 6 144 400 180 400 c is larger 10 11 1 100 11 144 1 144 a +b is larger Obtuse Triangle Acute Triangle 8) 1, 1, 170 9) 5, 4 5, 4 Figure out which is the largest side Figure out which is the largest side 1 1 1 169 170 170 170 is the c 1 1 170 1169 170 170 170 Right Triangle 5 0 4 5 80 4 48 4 5 is the c 5 4 4 5 0 48 80 68 80 a +b is larger Acute Triangle
Inequalities in One Triangles Objectives: SWBAT use triangle measurements to decide which side is longest or which angle is largest. SWBAT use the triangle inequality. SWBAT Recognize side inequalities of a Triangle Comparing Triangles vs. Angle Theorem 1. The smallest angle is opposite the smallest side. The smallest angle is opposite the smallest side Examples Name the shortest and longest sides of the triangle. 1.. 6 48 mm 180 Smallest Angle 6 mm 84 180 Medium Angle 48 mm 96 Larg e Angle 96 Smallest KM Medium LM Larg est KL 6 68 mo 180 Smallest Angle 49 mm 11 180 Medium Angle 6 mm 49 Larg e Angle 68 Smallest MN Medium ON Larg est MO
Name the smallest and largest angles of the triangle.. 4. E 9 F H 1 11 5 G Smallest Angle G Medium Angle E Larg e Angle F 6 I J Smallest Angle H Medium Angle I Larg e Angle J List all the angles and sides in ascending order. 5. 6. 90 45 mp 180 mp 15 180 mp45 Smallest Angle M TIED Smallest Angle P Larg e Angle N Smallest MN TIED Smallest NP Larg est MP ma mb mc 180 x x 6 5x 180 10x 0 180 10x 150 ma 15 47 x 15 mb 15 6 56 mc 515 77 Smallest Angle A Medium Angle B Smallest BC Larg e Angle C Medium AC Larg est AB
Triangle Inequality Theorem 1. The smallest side must be larger than the difference of the other two sides. The largest side must be the smaller than the sum of the other two sides. Smallest Larg est Subtract s Add s. If the triangles does not fulfill both stipulations it can t be a triangle. Is it possible to have the following dimensions for triangle ABC? Add two sides together, must be larger than the rd side (do this times) 7. AB = 8, BC = 15, and AC = 17 8. AB = 6, BC = 8, and AC = 14 AB BC AC 8 15 17 17 AB AC BC 8 17 15 5 15 AC BC AB 15 17 6 6 Since all checks work then it is a triangle. AB BC AC 6 8 14 14 14 Can' t Equal Since this doesn t work, it can t be a triangle. 9. AB = 1, BC = 1, and AC = 10. AB = 5, BC = 5, and AC = 5 AB BC AC 11 Can' t Equal AB BC AC 5 5 10 10 5 Since this doesn t work, it can t be a triangle. AB AC BC 5 5 5 10 5 AC BC AB 5 5 5 10 5 Since all checks work then it is a triangle.
Find the possible measures for XY in XYZ. 11. XZ = and YZ = 1. XZ = 8 and YZ = 10 1. XZ = 7 and YZ = 11 Smallest Subtract 1 s Smallest Subtract 10 8 s Smallest Subtract 117 4 s Larg est Add 5 s Larg est Add 10 8 18 s Larg est Add 117 18 s Smallest L arg est 1 5 Answer : 1 x 5 14. XZ = 1 and YZ = 15 Smallest L arg est Smallest L arg est 10 Answer : 4 18 x 10 Answer : 4 x 18 Smallest Subtract 15 1 s Larg est Add 15 1 7 s Smallest L arg est 7 Answer : x 7
Special Right Triangles Day 1 0 60 90 Right Triangles Objectives: SWBAT find the side lengths of special right triangles. Review: Rationalizing the Denominator Can t have a root in the denominator so fix this you need to rationalize Multiply the numerator and denominator by the root and simplify 1.. 5. 6 4. 5 6 4 5 5 5 9 5 6 6 6 4 6 6 4 Re duce 5 6 5 6 6 6 5 1 6 5 6 10 6 5
Write the sides and angles from shortest to longest. 1.. 90 60 mp 150 mp 150 150 mp0 s Angles s Angles Smallest MN Smallest Angle P Smallest DB Smallest Angle C Medium NP Medium Angle M Medium CD Medium Angle B Larg est PM Larg e Angle N Larg est CB Larg e Angle D So in a right triangle, the largest angle or right angle will always be opposite the Largest or Hypotenuse. The smallest angle will always be opposite from the smallest side. So in a Special 0 60 90 Right Triangle, Label the sides (short, medium / middle, and hypotenuse) by their length given the following angle measurements.
Rules for the 0-60 - 90 Special Right Triangle If you are getting Larger than you _Multiply. x 60 x If you are getting Smaller than you Divide. 0 x What I Have. What I want. What I do. SHORT LEG MIDDLE / LONG LEG HYPOTENUSE Middle / Long Leg Multiply by Hypotenuse Multiply by Short Leg Divide by Hypotenuse Find Short Leg First then find the Hypotenuse Short Leg Divide by Middle / Long Leg Find Short Leg First then find the Middle Leg Examples: Find the short leg given the hypo. 1.. Hypotenuse What I Have: Small Leg What I Need: Getting: Smaller / Bigger (Circle) Hypo Small 16 8 Hypotenuse What I Have: Small Leg What I Need: Getting: Smaller / Bigger (Circle) Hypo Small 1 6
Find the hypo given the short leg.. 4. Hypotenuse What I Have: Small Leg What I Need: Getting: Smaller / Bigger (Circle) Small Hypo 17 4 Find the middle leg given the short leg. Hypotenuse What I Have: Small Leg What I Need: Getting: Smaller / Bigger (Circle) Small Hypo 10 0 5. 6 5 Small Leg What I Have: Middle / Long Leg What I Need: Getting: Smaller / Bigger (Circle) Small Middle / Long 11 11 Small Leg What I Have: Middle / Long Leg What I Need: Getting: Smaller / Bigger (Circle) Small Middle / Long 5 5 9 5 15
Find the short leg given the middle leg. 7. 8. 15 Middle / Long Leg What I Have: Small Leg What I Need: Getting: Smaller / Bigger (Circle) Middle / Long Small 15 15 15 Middle / Long Leg What I Have: Small Leg What I Need: Getting: Smaller / Bigger (Circle) Hypo Small 6 6 6 6 Re duce Outsides
Identify the name of the leg, then solve for the variable in the following Right Triangles. 9. 10. Small Leg What I Have: Middle / Long Leg What I Need: Getting: Smaller / Bigger (Circle) Small Middle / Long 1 1 4 What I Have: Middle / Long Leg Small Leg What I Need: Getting: Smaller / Bigger (Circle) Middle / Long Small 4 4 11. 1. 4 10 Middle to Hypo Find the Short Leg First Hypo to Middle Find the Short Leg First Middle / Long Short 10 10 Small Leg 10 Short Hypo 10 0 Hypo Short 1 6 Short Leg 6 Short Middle / Long 6 6
1. Middle / Long Short 9 9 9 9 9 9 Small Leg Short Hypo 6 14. The size of a television is determined by the diagonal measure of its rectangular screen. The figure below represents a television screen. What is the size of the TV? Round to the nearest inch. Looking for the Hypo Given the Middle Leg Find the Small Leg First Middle / Long Short 45 45 45 45 9 45 15 Small Leg 15 Short Hypo 15 0 5in
Rationalizing the Denominator Special Right Triangles Day Objectives: SWBAT find the side lengths of special right triangles. Can t have a root in the denominator so fix this you need to rationalize Multiply the numerator and denominator by the root and simplify Examples: 5 1. 4. 4. 4. 5 Root 4 Can be Broken Down 4 5 5 4 5 4 5 6 9 4 9 4 5 6 Re duce Re duce 1
Rules for the 45-45 - 90 Special Right Triangle If you are getting Larger than you _Multiply. x 45 x If you are getting Smaller than you Divide. x 45 What I Have. What I want. What I do. LEG HYPO Multiply by Hypotenuse LEG Divide by Find the hypo given the leg. 1.. Getting: Smaller / Bigger (Circle) Getting: Smaller / Bigger (Circle) LEG HYPO LEG HYPO 11 11
Find the leg given the hypo.. 4. 7 Getting: Smaller / Bigger (Circle) Getting: Smaller / Bigger (Circle) Hypo Leg 7 7 7 Examples: Solve for the variable in the following Right Triangles. 5. 6. Hypo Leg 16 16 16 16 4 16 Re duce 8 Getting: Smaller / Bigger (Circle) LEG HYPO 1 1 Getting: Smaller / Bigger (Circle) LEG HYPO 6 6
7. 8. Hypo Leg 5 5 5 5 4 5 Hypo Leg 6 6 6 6 4 6 Re duce 9. Given the following quadrilateral is a rectangle. y 45 z 4 Leg Hypo 4 w 4
10. Jason is adding a climbing wall to tis little brother s swing set. If he starts building 5 feet out from the existing structure, and wants to have a 45 degree, how long should the wall be (round to the nearest tenth)? LEG HYPO 5 5 71. ft
Trigonometric Ratios Day 1 Objectives: SWBAT set up the fractions for sine, the cosine and the tangent Trigonometric Ratios: Sine The Ratio in a Triangle Opposite Hypotenuse Cosine The Ratio in a Triangle Adjacent Hypotenuse Tangent The Ratio in a Triangle Opposite Adjacent Θ Greek Letter Theta / Angle you will be using 1.. A) List the angles from smallest to largest. A) List the angles from smallest to largest. Smallest Angle A Can' t Tell Without Numbers Medium Angle C Larg e Angle B DO NOT ASSUME B) What is the cosine of angle A? B) What is the tangent of angle M? cos A Adjacent Hypo 15 17 tan A Opposite m Adjacent n
Find the sine, the cosine, and the tangent. 4. R. S Sine: sin S Opposite Hypo 5 1 Sine: sin R Opposite Hypo 1 1 Cosine: cos S Adjacent Hypo 1 1 Cosine: cos R Adjacent Hypo 5 1 Tangent: tan S Opposite Adjacent 5 1 Tangent: tan R Opposite Adjacent 1 5 Where are the fractions exactly the same? Sine and Cosine of the other acute angles will match 5. Set up the following fractions for each trig relationship. You must use Pythagorean Theorem to find the missing rd side of the right triangle. a b c A. sin(θ) = sin S Opposite Hypo 5 b 5 9 b 5 b 16 b 4 B. cos(θ) = cos S Adjacent Hypo 4 5 C. tan(θ) = tan S Opposite Adjacent 4 4
Trigonometric Ratios Day II Objectives: SWBAT find the sine, the cosine and the tangent of an acute angle. Trigonometric Ratios: Sine The Ratio in a Triangle Opposite Hypotenuse Cosine The Ratio in a Triangle Adjacent Hypotenuse Tangent The Ratio in a Triangle Opposite Adjacent Degree Mode of a Calculator Most calculator have multiple modes for handling sine, cosine, and tangent (based on the type of math and information). Make sure you calculator is in DEG mode Use a calculator to approximate the measure value. (round to nearest thousandth) 1. sin 0 =. tan 5 4 =. cos 77 = 4. sin 5 14 = 0.500 0.0 0.5 0.006
For each example, answer the following. Round to the nearest thousandth. 5. 6. A) Will BC be larger or smaller than AB? Why? A) Which will be larger NP or PM? Why? Larger because A is larger PM because N is larger than M B) Find x(round to the nearest thousandth). B) Find y (Round decimal places). sin A Opposite Hypo x sin51 15 sin51 x 1 15 Cross Multiply x 15sin 51 1 15 0. 77711x x 11. 657 cos M Adjacent Hypo y cos8 14 cos8 y 1 14 Cross Multiply y 14 cos 8 1 14 0. 7880 1y y 11. 0
7. 8. A) Will AC be larger or smaller than AB? Why? A) What side will be the longest side? Why? Smaller because C is larger TR because it is opposite the Right Angle B) Find z (Round to the nearest thousandth). B) Find y (Round decimal places). tan C Opposite Adjacent z tan55 8 tan55 z 1 8 Cross Multiply z 8 tan 55 1 8 1. 48 1z z 11. 45 cos T cos Adjacent Hypo 51 8 y cos 51 8 1 y Cross Multiply y cos 51 18 Trig Functions are Just Decimals y 0. 69 8 0. 69y 8 0. 69y 8 0. 69 y 1. 71
9. A) What will the largest side be in ABC? Why? The largest side will be AC because it is opposite the Right Angle B) What will the shortest side be in ABC? Why? BC will be the smallest because A is opposite the smallest angle C) Find the value of x (Round to the nearest thousandth). sin 9 10. Solve for x in the diagram below. Round to the nearest Thousandth. Can t Find x without Another side to that triangle Opposite Hypo x sin9 15 sin9 x 1 15 Cross Multiply x 15 sin 9 1 15 0. 69 1x x 9. 440 a b c 11 y 1 11 y 441 y 0 y 17. 889 cos A Adjacent Hypo 17. 889 cos4 x cos 4 17. 889 1 x Cross Multiply 4 117 889 x cos. Trig Functions are Just Decimals y 0. 71 17. 889 0. 71y 17. 889 0. 71y 17. 889 0. 71 y 4. 460
11. A 4 foot ladder is leaned at a 70 degree angle against a building. How far is the base of the ladder from the building? cos A Adjacent Hypo x cos70 4 cos70 y 1 4 Cross Multiply y 4 cos 70 1 4 0. 4 1y y 8. 08 ft
Solving Right Triangles Inverses of Sine, Cosine, and Tangent Objectives: SWBAT solve a right triangle (Write just like sine, cosine, tangent if you are looking for an angle, switch the fraction and the angle and put in the inverse sign) Sin(A) = x or a fraction 1 sin ( x) ma Cos (A) = x or a fraction Tan (A) = x or a fraction 1 cos ( x) ma 1 tan ( x) ma Number of Degrees in a Triangle 180 Degrees in a Triangle Examples: Write the correct ratio for each trig function. 5 1 1. sin A =. cos A = 5 1. tan A = 4. sin B = 5 1 5. cos B = 6. tan B = 1 1 1 1 1 5 7. cos C = Can t Do 8. sin C = Can t Do 9. tan C = Can t Do You can t do any trig functions from C or 90 degree angles.
Use a calculator to approximate the measure of angle A. (round to nearest thousandth) 1. sin A = 0.4. tan A = 5 sin 0.4 1 ma tan.5 1 ma.85 ma 68.0 ma. cos A = 0.98 4. cos A = 17 cos 0.98 1 ma cos 0.68 1 5 ma 11.48 ma 47.156 ma Use the diagram to find the indicated measurement. 5. CR 6. m T Pythagorean Theorem for last side Trig ratios to find one of the Angles a b c a a 8 1 64 144 64 64 a 80 a 4 5 or 8.944 cos cos 1 1 8 m B 1 m B 48.190 mb 1 C 7. m C T 8 R Use 180 Degrees in a Triangle to find the last angle 180 mt mc mr 180 mt 48.190 90 180 mt 11.81 18.19 18.19 41.810 mt
Solve the right triangle (find all angles and sides) 8. Pythagorean Theorem for last side Trig ratios to find one of the Angles a b c a a 4 9 16 81 16 16 a 65 a 65 or 8.06 sin m A 9 1 4 6.88 ma Use 180 Degrees in a Triangle to find the last angle 180 mc ma mb 180 mc 6.88 90 180 mc116.88 116.88 116.88 6.61 mc
Solve the right triangle (find all angles and sides) R 9. 10. M 8 L S 11 T 14 8 N Pythagorean Theorem for last side Use 180 degree theorem 1 st on # a b c 180 mm mn ml 8 11 c 180 90 8 ml 64 11 c 180 ml118 c 185 118 118 c 185 6 ml Trig ratios to find one of the Angles tan m T 11 6.0 mt 1 8 Use 180 Degree theorem to finish the Angles 180 mr ms mt 180 mr 90 6.0 180 mr16.0 16.0 16.0 5.97 mr Use Trig Function to find a side ML sin 8 14 14sin 8 ML 6.57 ML Use Pythagorean to find last side a b c a a a 6.57 14 4.17 196 4.17 4.17 15.8 a 1.6
Solving Right Triangles If you need to find a side use If you need to find an angle use Pythagorean Theorem 180 Degrees in a Triangle Trig Functions Inverse Trig Functions Special Right Triangles
Trigonometric Ratios Application Objectives: SWBAT apply Trig Functions to Real Life Scenarios Angle of Elevation Looking up from the horizon Angle of Depression Looking Down from the horizon 1. A six foot person (measuring from his eye level to the ground) is looking up at a tree. How tall is the tree? y tan48 10 tan48 y 1 10 Cross Multiply y 10 tan 48 1 y 11. 106 ft Height of the Tree y 6 ft Person 11. 106 6 17. 106 ft
. Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 5 degrees, what is the horizontal distance from the seal to the cliff? tan 5 40 y tan 5 40 1 y Cross Multiply y tan 5 140 y 1. 51 ft. At the circus, a person in the audience at ground level watches the high-wire routine. A 6 foot tall acrobat is standing on a platform that is 5 feet above the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member s line of sight to the top of the acrobat s head is 7 degrees? 1 sin7 x sin7 1 1 x Cross Multiply x sin 7 11 y 68. 8 ft
4. Two buildings, of different sizes, are on opposite sides of a street. The street is 60 feet long. The taller building is 100 ft. tall, and the angle of depression from the top of the taller building to the top of the shorter building is 6.565. Find the height of the shorter building. y tan 6. 565 60 tan 6. 565 y 1 60 Cross Multiply 60 tan 6. 565 1 y y 0. 000 ft Smaller Building Tall Small 100 0 100 0 70 ft 5. Dante is standing at ground level x feet from the base of the Empire State Building in New York City. The angle of elevation formed from the ground to the top of the building is 48. 4. The height of the Empire State Building is 147 feet. What his distance from the Empire State Building to the nearest foot? 147 tan 48. 4 x tan 48. 4 147 1 x Cross Multiply x tan. 48 4 1147 y 106. 90 ft
6. A person is 6.5 feet tall and has a shadow for feet. What is the angle of elevation to the sun at the time? 6.5 tan x 1 6.5 tan x 65.5 x