Inverse Trig Functions - Classwork

Transcription

1 Inverse Trig Functions - Classwork The left hand graph below shows how the population of a certain city may grow as a function of time. <f you are interested in finding the time at which the population reaches a certain value, it may be more convenient to reverse the variables and write time as a function of population. The relation you get by interchanging the two variables is called the inverse of the original function. The graph of the inverse is shown on the right graph below. Population Time Inverse relation y= Original Function Time Population For a linear function such as A 6, interchanging the variables y A 6 for the inverse relation. Solving for y in terms of gives 0.D ". The symbol f, pronounced Ff inverse,g is used for the inverse = + = function. <f f 6, then f <f f turns out to be a function Hpasses the vertical line testi, the the original function f is said to be invertible. Jemember that the " eponent does not mean the reciprocal of f ( ). The inverse of a function undoes what the ( ) = function did to. That is f f. <f f =, then f ( ) = and =. Note that if the same scales are used for the two aes, then the graphs of f and f are mirror images with respect to the 4D o line The inverses of the trigonometric functions follow from the definition. For instance, if the function is y = sin, the inverse function is given by = sin y. When we solve for y, we get y = sin. The symbol arcsin is sometimes used to help you distinguish sin from. Oere are the graphs of y = sin sin and y = sin. <t is obvious that the inverse sine relation is not a function. There are many values of y for the same value of. To create a function that is the inverse of sin it is customary to restrict the range to " % y % ". This includes only the branch of the graph nearest the origin. Hrd picture abovei. MasterMathMentor.com " 0 " Stu Schwartz

2 Pelow are pictures of the inverse cosine function and inverse tangent function. Note that in order to ensure that these relations are functions, we have to restrict the range. So, we have these definitions: $ y = y = y # " " ' sin if and only if sin and &, ) % ( y = cos if and only if cos y = and y # 0, " y = y = y # " " - tan if and only if tan and,, +. You must Snow conversions of degrees to radians and special triangles. 60 o or 80 o. Some of the relationships that you should Snow are: Degrees 0 4D D D0 80 Radians D <n a 0 o "60 o "90 o triangle, the sides are always in the proportion " 0 ". <n a 4D o "4D o "90 o triangle, the sides are always in the proportion " " 0. & ' Eample I "Evaluate each of the following: ai arcsin, + -. b. cos 0 c. tan d. csc ( ) MasterMathMentor.com " 0 " Stu Schwartz

3 Eample I Evaluate the following. MaSe a picture to describe the situation. a. sin arctan - -, b. tan arccos + 4., +. c. sec, sin d. cot, cos + 5. Eample I Evaluate the following. MaSe a picture to describe the situation. ( ) a. cos sin b. tan cos c. sin( cos ) d. sin tan So now we can tase derivatives of inverse trig functions. Find d sin y = sin sin y =. y = sin sin y = Draw a picture y the angle is y, opposite =, hypotenuse = Remaining side is Since sin y =, tase the derivative of each side cos y dy dy dy = or = or = cos y Eample 4I TaSe the derivative of a. y = cos b. y = tan MasterMathMentor.com " 0 " Stu Schwartz

4 The derivatives of the three inverse trig functions are as follows: d ( sin u) = u d ( cos u) = u d tan u u = + Eample DI Find the derivatives of a. y = sin 4 b. y = tan du du du c. y = +, - cos. d. y = sin + Eample 6I Xn officer in a patrol car sitting 00 feet from the highway observes a trucs approaching. Xt a particular instant t seconds, the trucs is feet down the road. The line of sight to the trucs mases an angle of 4 radians to a perpendicular line to the road. Truck a. Epress 4 as an inverse trig function. 00 ft 4 b. Find d 4 dt Patrol Car c. When the trucs is D00 ft, the angle is observed to be changing at a rate " degreesysec. Oow fast is the car going in ftysec and mphz dt MasterMathMentor.com " 04 " Stu Schwartz

5 Inverse Trig Functions - Homework I "Evaluate each of the following: a. arccos b. cot c. sin d. sec I Evaluate the following. MaSe a picture to describe the situation. a. cos arcsin -, b. sin arctan -, c. csc( cot ) 4 d. tan csc I Evaluate the following. MaSe a picture to describe the situation. ( ) a. cos tan b. sec sin c. tan( sin ) ( ) 4 d. cos tan + 4I Find the derivatives of a. y cos b. y = sin = MasterMathMentor.com " 0D " Stu Schwartz

6 c. y tan 5 d. y cos 0 = = e. y = arctan f. y sin cos t = DI Find any relative etrema of y = arcsin 6I The base of a 0 foot tall eit sign is 0 feet above the driver[s eye level. When cars are far away, the sign is hard to read because of the distance. When they are close, the sign is hard to read because the driver has to loos up at a steep angle. The sign is easiest to read when the distance is such that the angle 4 at the driver[s eye is as large as possible. a) Write 4 as the difference of inverse tangents. 0 ft Eit Mile ahead b) Write an equation for d4 4 0 ft Car c) The sign is easiest to read at the value of where 4 stops increasing and starts decreasing. This happens when d 4 = 0. Find and confirm using the calculator. MasterMathMentor.com " 06 " Stu Schwartz