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Lecture 16 Right Triangle Trigonometry 55º 8 feet 5 ft How high is Sparty? 1 © m j winter, 2000 All Right Triangles with a 55º Angle Are Similar C” C’ C A 55o 55o 55o A B CB/AB = C’B’/AB’ B” A B’ = C”B”/AB” = tan55 o 2 1 Basic Trigonometric Functions (Tri -Gon = 3 sides) b opposite = c hypotenuse a adjacent cos(θ ) = = c hypotenuse sin(θ ) = tan(θ ) = b opposite = a adjacent b = opposite c = hypotenuse q a = adjacent 3 All Right Triangles with a 55º Angle Are Similar tan(55º) = 1.42814... 55º 8 feet y 8 55 o 5 ft 5 y tan(55o )=1.42814 8= y = 8(1.42814) =11.425 Sparty = y+ 5 = 16.425 4 2 Values of Trig. Functions Two cases when exact answers are used: 30o-60o-90o triangle and 45 o-45o-90o triangle s 30 s s h 60 o sin60 h 2 h2 = s 2 − h=s o s/2 s h2 + = s 2 2 2 2 s 3s = 4 4 3 2 3 2 = 3 s 2 s 1 cos60 = 2 = s 2 60 s = s 3 2 tan60 = = 3 s 2 1 sin30o = cos60o = 2 o Notice that s does not appear in the values of the functions s cos30o = sin60o = 3 2 5 45o-45o-90o triangle 2 2 s +s =c 2 sin45o = cos45o = 2 s 2 = c2 s 1 2 = = 2 s 2 2 s 2 =c 45 For all other angles, use a calculator. c s Make sure it is in degree mode! 45 s 6 3 How to Remember These θ Sin( θ) 0 o 30 o 45 o 60 o 90 Cos(θ) 0 2 1 1 = 2 2 2 2 3 2 4 =1 2 4 =1 2 3 2 2 2 1 1 = 2 2 0 2 Tan(θ) 0 1 3 1 3 Not defined 7 Applications - 1 Angle of elevation Angle of Depression If you are sitting 20 feet up in a tree stand and see a deer at an angle of depression of 42 degrees, how far is the deer from the base of the tree? How far is the deer from you? Assume the tree is vertical. you 42 o 20’ deer 8 4 Solution to Deer problem you x = tan48o 20 42 o 48 y 20’ x = 20 ⋅ tan48o = 22.2122feet deer x Can find y by using trigonometry or by using Pythagorean theorem x 2 + 202 = y2 or 20/y = cos(48o) y = 29.89 feet 9 Piccadilly Circus The traffic at Piccadilly Circus is so bad that you cannot get very close to the statue in the middle. From a spot on a side street, you look at the statue and note that the angle of elevation is about 45º. You walk about 20 feet closer and observe that the angle has now become 60º. How high is the statue? 10 5 PC - diagrams Firstly, make a diagram and label all known quantities Second: label unknown parts h 45º 60 45º 20 20 60 x 11 45º 20 60 x PC - analysis - Two triangles h h = tan60o x h = tan45o 20 + x h 45º 20 We want to find h, so we’ll solve both equations for x and equate them. 60 x h 45º 20 60 12 x 6 h = tan60o x h = tan45o 20 + x PC - analysis - Two triangles h h x= = 3 tan60o h 20 + x = tan45o x = h − 20 = h 1 Solve for h: h = h − 20 3 13 PC - finish h = h − 20 3 3 − 1 1 20 = h 1− = h 3 3 h= 20 3 = 47.32 3 −1 Statue is 47.32 feet higher than your head. 14 7 Inverse Trig Functions How do we solve these: cos(x) = .456 sin(x) = .321 tan(x) = 1.5 15 Inverse Trig Functions To solve: cos(x) = .456 sin(x) = .321 tan(x) = 1.5 Use the inverse trig functions! Old fashioned language arccos(x) means “the angle whose cosine is x” arcsin(x) means “the angle whose sine is x” arctan(x) means “the angle whose tangent is x” 16 8 Inverse Trig Functions - 2 To solve: cos(x) = .456 sin(x) = .321 tan(x) = 1.5 Modern usage: arccos(x) = “the angle whose cosine is x” becomes cos–1(x), the inverse cosine of x. arcsin(x) = “the angle whose sine is x” becomes sin–1(x), the inverse sine of x. arctan(x) = “the angle whose tangent is x” becomes tan–1(x), the inverse tangent of x. 17 Inverse Trig Functions - 3 To solve: cos(x) = .456 sin(x) = .621 tan(x) = 1.5 Modern usage: arccos(x) = “the angle whose cosine is x” becomes cos–1(x), the inverse cosine of x. arcsin(x) = “the angle whose sine is x” becomes sin–1(x), the inverse sine of x. arctan(x) = “the angle whose tangent is x” becomes tan–1(x), the inverse tangent of x. cos–1(.456) = 62.87 o sin–1(.621) = 38.39o tan–1(1.5) = 56.31o 18 9 Pythagoras and Trigonometry c a θ b a2 + b 2 = c2 a2 + b 2 = 1 c2 c2 2 2 a b =1 c + c (sin(θ)) 2 + (cos(θ))2 = 1 sin 2(θ) + cos 2(θ) = 1 19 10