6.6 – TRIG INVERSES AND THEIR GRAPHS Pre-Calc

Inverse Trig Functions REVIEW SLIDE Inverse Trig Functions Original Function Inverse y = sin x y = sin-1 x y = arcsin x y = cos x y = cos-1 x y = arccos x y = tan x y = tan-1 x y = arctan x

Consider the graph of y = sin x REVIEW SLIDE Consider the graph of y = sin x What is the domain and range of sin x? What would the graph of y = arcsin x look like? What is the domain and range of arcsin x? Domain: all real numbers Range: [-1, 1] Domain: [-1, 1] Range: all real numbers

Is the inverse of sin x a function? REVIEW SLIDE Is the inverse of sin x a function? This will also be true for cosine and tangent. Therefore all of the domains are restricted in order for the inverses to be functions.

Original Functions with Restricted Domain REVIEW SLIDE How do you know if the domain is restricted for the original functions? Capital letters are used to distinguish when the function’s domain is restricted. Original Functions with Restricted Domain Inverse Function y = Sin x y = Sin-1 x y = Arcsin x y = Cos x y = Cos-1 x y = Arccos x y = Tan x y = Tan-1 x y = Arctan x

Original Domains  Restricted Domains REVIEW SLIDE Original Domains  Restricted Domains Domain Range y = sin x all real numbers y = Sin x y = cos x y = Cos x y = tan x all real numbers except n, where n is an odd integer y = Tan x

Sketch a graph of y = Sin x Remember principal values Create a table

Now use your table to generate Sin-1 IF YOU CAN REMEMBER AND MEMORIZE WHAT THE original and inverse funcitons look like, it will make your life a lot easier!!!

Table of Values of Cos x and Arccos x y = Cos x X Y 1 π/3 0.5 π/2 2π/3 -0.5 π -1 y = Arccos x X Y 1 0.5 π/3 π/2 -0.5 2π/3 -1 π

The other trig functions require similar restrictions on their domains in order to generate an inverse. Like the sine function, the domain of the section of the tangent that generates the arctan is Y=Arctan(x) Y=Tan(x)

Table of Values of Tan x and Arctan x y = Tan x X Y -π/2 UD -π/4 -1 π/4 1 π/2 y = Arctan x X Y UD -π/2 -1 -π/4 1 π/4 π/2

Write an equation for the inverse of y = Arctan ½x Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse. To write the equation: Exchange x and y Solve for y Let’s graph 2Tan x = y first. Complete the table: Then graph! x = Arctan ½y Tan x = ½y 2Tan x = y 1 y = 2Tan x X Y -π/2 Undefined -π/4 -2 π/4 2 π/2 Now graph the original function, y = Arctan ½x by switching the table you just completed! π/2 -π/2 -1

Now graph the original function, y = Arctan ½x by switching the table you just completed! y = 2Tan x X Y -π/2 UD -π/4 -2 π/4 2 π/2 y = Arctan ½x X Y UD -π/2 -2 -π/4 2 π/4 π/2 π/2 1 -1 -π/2

Write an equation for the inverse of y = Sin(2x) Write an equation for the inverse of y = Sin(2x). Then graph the function and its inverse. y = Sin2x X Y -π/4 -1 -π/12 -.5 π/12 .5 π/4 1 Let’s graph y = Sin(2x) first. The domain changes because of the 2, how? To write the equation: Exchange x and y Solve for y Divide all sides by 2 x = Sin(2y) Arcsin(x) = 2y ½Arcsin(x) = y 1 y = ½Arcsin(x) X Y -1 -π/4 -.5 -π/12 .5 π/12 1 π/4 Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed! -π/2 π/2 π -1 Remember you can always check and see if they are symmetric with respect to y = x

Graph the inverse of: Let’s find the inverse equation first: X Y -1 -1 π/4 π/2 3π/4 π 1 Flip the “x” and “y” and solve for “y”: Take the sine of both sides 1 Domain is now: π/2 π Add π/2 to all three sides -1

Graph What is the domain for Sin(x)? Since we are graphing Arcsin the domain will become the range, but it will change!! Solve for x: y = (π/2)+Arcsin x X Y -1 π/4 π/2 3π/4 1 π Now make a table using the y-values as your input into this function: Take the sine of both sides π y = Arcsinx Domain is now: π/2 Add π/2 to all three sides -1 1 Just shifted up π/2

Now try to graph just by using the shifting technique. π y = Arcsinx π/2 Just shifted down π/4 -1 1 -π/2

Now try to graph just by using the shifting technique. Just shifted up π/4 π y = Arctan(x) π/2 -1 1 -π/2

Graph:

Determine if each of the following is true or false Determine if each of the following is true or false. If false give a counter example. Cos-1(cos x) = x for all values of x Sin-1(sin x) = x for all values of x Sin-1x + Cos-1x = π/2 -1 ≤ x ≤ 1 Arccos x = Arccos (-x) -1 ≤ x ≤ 1 Tan-1x = 1/(Tan x) x = 270° Cos-1(cos 270°) = Cos-1(0) = 90° FALSE x = 180° or try x = 225° Sin-1(sin 180°) = Sin-1(0) = 0° FALSE x = 0 of try x = 1 or -1 Sin-1(0) + Cos-1(0) = 0° + 90°= 90° TRUE x = 1 or try x = -1 Arccos(1) ≠ Arccos (-1) 0° ≠ 180° FALSE x = 0 Tan-1 (0) ≠ 1 / (Tan (0)) 0° ≠ 1 / 0 UNDEFINDED FALSE

Evaluate each expression -30 degrees 45 degrees

Evaluate each expression 1 Negative square root of 3