C HAPTER 9.10 TRIGONOMETRIC RATIOS By: Arielle Green Mod 9

Sin = Cos = Tan =

V OCABULARY Angle of Elevation – the angle between an upward line of sight and the horizontal is the angle of elevation. Horizontal line Line of sight C A B angle of elevation

S AMPLE P ROBLEM 1 A girl was walking in the woods when she stopped 10 ft away from a tree. She spotted a birds nest at an angle of elevation of 37˚. How far up from the ground was the birds nest rounded to the nearest tenth? 10 37˚ X First choose the formula needed for this problem. We are working with the two legs of the right triangle, so we will use tan. Set up the formula and solve for x. R S ft Q

V OCABULARY Angle of Depression - the angle between a downward line of sight and the horizontal is the angle of depression. X Y Z line of sight horizontal line W Angle of depression

An airplane pilot is flying over a forest at an altitude of 1600 ft. Suddenly, he spots a fire. He measures the angle of depression and finds it to be 46˚. How far is the fire, rounded to the nearest tenth, from a point on land directly below the plane? There are two ways to solve this problem. We’ll look at both ways.

1600 X A B C 46˚ Using parallel lines alt. int., <ACB is also 46˚. Since only the two legs of the right triangle are being used, the formula must be Tan =. Set up the equation and solve for x. Tan 46 = x = x ft D

A BC ˚ 44˚ X D Since <BAC and <CAD are complementary <s, <BAC is 44˚. Only the two legs of the right triangle are being used, so the formula must be Tan =. Set up the equation and solve for x. Tan 44 = X = 1600 ∙ tan 44 x ft

P RACTICE PROBLEMS R OUND ALL ANSWERS TO THE NEAREST TENTH. R OUND ALL ANGLES TO THE NEAREST DEGREE. 1.) A lighthouse casts a shadow of 55 ft when the sun is at an angle of elevation of 67 ˚. How tall is the lighthouse? 2.) A cat was on a cliff when it saw a mouse down below at an angle of depression of 25 ˚. The cliff is 43 ft tall. How far away is the mouse from the bottom of the cliff? 3.)A 25-foot ladder just reaches a point on a wall 24 ft above the ground. What is the angle of elevation of the ladder?

P RACTICE PROBLEMS Two men are on the opposite sides of a tall building with the angle of elevation being 30 and 60 respectively. If the one man is 40 feet away from the base of the building, how far away is the other man? A pole 40 ft high has a shadow the length of 23 ft at this point in time. Find the angle of elevation of the sun. Harry was walking along a pier. He stopped when he saw a boat on the lake at an angle of depression of 22˚. If the boat is 65 ft away, how high, rounded to the nearest tenth, is the pier from the water ? 4.)5.) 6.) x 30˚60˚ 40

A NSWERS TO THE PRACTICE PROBLEMS 1.) 2.) 3.) x ˚ ˚ x X˚X˚ ¯¹

A NSWERS TO PRACTICE PROBLEMS ( CONT ’ D ) 4.)5.) 8 60˚30˚ x ft x˚x˚ ¯¹ ˚ 6.) 68 ˚ 65 x ft 22 ˚

W ORKS C ITED Rhoad,Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal Little, “Math:Trigonometry.”Syvum Syvum technologies. 29 May