Lesson 9-R Chapter 8 Review

Objectives Review Chapter 9 Material in preparation for the test

Vocabulary None new

Geometric and Arithmetic Means Arithmetic Mean (AM) or average of 2 numbers: (a + b) / 2 Geometric Mean (GM) of 2 numbers: √ab Altitude Length = GM of divided hypotenuse = √ab a b altitude

Special Right Triangles b c Pythagorean Theorem: a2 + b2 = c2 Side opposite 30° angle is ½ the hypotenuse Side opposite 45° angle is ½ the hypotenuse times √2 Side opposite 60° angle is ½ the hypotenuse times √3 Pythagorean Triples: Whole numbers that solve the theorem (example: 3,4,5) 45° 30° 60° x y ½ x√2 ½y ½ y√3 ½ x√2

Trigonometric Functions Sin (angle) = Opposite / Hypotenuse Cos (angle) = Adjacent / Hypotenuse Tan (angle) = Opposite / Adjacent SOH – CAH – TOA (or others) to help remember the definitions To find an angle use the inverse of the Trig Function Trig Fnc-1 (some side / some other side) = angle Remember the shortcut for the bottom of a fraction hypotenuse opposite angle adjacent 8 8 --- = 0.781  --------- = x just switch x and the = # x 0.781

Trig Problems Steps to Solution Step 1: Label sides (A, H, O) based on angle Step 2: Identify trig function to use Step 3: Set up equation Step 4: Solve for variable (1 of these methods) if variable is in top of fraction, multiply both sides by the bottom to get “x = …” if variable is in bottom of fraction, x trades places with what’s on the other side of the = sign to get “x = …” if variable is the angle, use inverse trig function notation to get “x = …” x sin 23° = ------- x = 45  sin 23° 45 21 21 cos 41° = ------- x = --------- x cos 41° 23 23 tan x° = ------- x = tan -1 ----- 37 37

Angles of Elevation or Depression To Solve: Step 1: Draw the triangle below Step 2: Label sides (A, H, O) from problem information Step 3: Identify trig function to use Step 4: Set up equation Step 5: Solve for variable (use 1 of the 3 methods) slant distance; ski slope or road vertical distance or height angle always goes here Θ horizontal distance or length of shadow

Summary & Homework Summary: Homework: Arithmetic mean is the average – (a+b)/2 Geometric mean Square root of the product -- √ab Length of the altitude (GM of divided hypotenuse) Pythagorean Theorem – a² + b² = c² Pythagorean Triples – whole numbers Special Case Right Triangles Side opposite 30° is ½ hypotenuse Side opposite 45° is ½ hypotenuse  √2 Side opposite 60° is ½ hypotenuse  √3 Trigonometric functions (SOH – CAH – TOA) Angle of Elevation or Depression Homework: study for the test

Problems 1. Find the Arithmetic Mean and the Geometric Mean of 3, 15 2. Find the altitude in the triangle to the right 3. Find the missing side in the triangle to the right 10 a 15 25 x 15

More Problems 4. Does 6, 8, 9 make a Right Triangle? A Pythagorean Triple? 5. Does 1, 4/3, 5/3 make a Rt Triangle? A Pythagorean Triple? 6. Solve for the variables in the triangle to the right 7. Solve for the variables in the triangle to the right 8. If a 20 ft ladder leans up against a barn at a 62° angle to the ground, how high up the barn does it reach? x 30° 26 y° z x 45° z y° 15 x 62° 20

Problems 1. Find the Arithmetic Mean and the Geometric Mean of 3, 15 2. Find the altitude in the triangle to the right 3. Find the missing side in the triangle to the right AM = (3+15)/2 = 18/2 = 9 GM = √ (3•15) = √45 = 6.71 10 a 15 GM = √ (10•15) = √150 = 12.25 25 x Pythagorean Theorem: hyp² = leg² + other leg² or c² = a² + b² 15 (25)² = (15)² + x² 625 = 225 + x² 400 = x² 20 = x

More Problems NO! 6² + 8² ≠ 9² NO! not a Rt ▲ 4. Does 6, 8, 9 make a Right Triangle? A Pythagorean Triple? 5. Does 1, 4/3, 5/3 make a Rt Triangle? A Pythagorean Triple? 6. Solve for the variables in the triangle to the right 7. Solve for the variables in the triangle to the right 8. If a 20 ft ladder leans up against a barn at a 62° angle to the ground, how high up the barn does it reach? Yes! (1)² + (4/3)² = (5/3)² NO! not all whole numbers x 30° 26 y° z By Trig: sin 30° = x / 26 0.5 = x / 26 13 = x cos 30° = z / 26 0.866 = z / 26 22.52 = z y = 90 – 30 = 60° x 45° z y° 15 By Trig: tan 45° = x / 15 1 = x / 15 15 = x cos 45° = 15 / z 0.707 = 15 / z 0.707z = 15 z = 21.22 y = 90 – 45 = 45° sin 62° = x / 20 0.883 = x / 20 17.66 = x x 62° 20 hyp opp