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6.6 – TRIG INVERSES AND THEIR GRAPHS Pre-Calc. Inverse Trig Functions Original FunctionInverse y = sin xy = sin -1 xy = arcsin x y = cos xy = cos -1 xy.

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Presentation on theme: "6.6 – TRIG INVERSES AND THEIR GRAPHS Pre-Calc. Inverse Trig Functions Original FunctionInverse y = sin xy = sin -1 xy = arcsin x y = cos xy = cos -1 xy."— Presentation transcript:

1 6.6 – TRIG INVERSES AND THEIR GRAPHS Pre-Calc

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

3 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 REVIEW SLIDE

4 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. REVIEW SLIDE

5 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 xy = Sin -1 xy = Arcsin x y = Cos xy = Cos -1 xy = Arccos x y = Tan xy = Tan -1 xy = Arctan x REVIEW SLIDE

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

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

8 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!!!

9 Table of Values of Cos x and Arccos x y = Cos x XY 01 π/3 0.5 π/2 0 2π/ π y = Arccos x XY π/3 0 π/ π/3 π

10 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=Tan(x) Y=Arctan(x)

11 Table of Values of Tan x and Arctan x y = Tan x XY - π/2 UD -π/4 00 π/4 1 π/2 UD y = Arctan x XY UD - π/2 -π/ π/4 UD π/2

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

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

14 Write an equation for the inverse of y = Sin(2x). Then graph the function and its inverse. To write the equation: 1.Exchange x and y 2.Solve for y x = Sin(2y ) Arcsin(x) = 2y ½Arcsin(x) = y Let’s graph y = Sin(2x) first. The domain changes because of the 2, how? Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed! y = Sin2x XY -π/4 -π/ π/12.5 π/4 1 1 π/2π -π/2 y = ½Arcsin(x) XY -π/ π/ π/12 1 π/4 Remember you can always check and see if they are symmetric with respect to y = x Divide all sides by 2

15 Graph the inverse of: Let’s find the inverse equation first: Flip the “x” and “y” and solve for “y”: XY 0 π/4 π /2 0 3π/4 π 1 Take the sine of both sides Domain is now: Add π/2 to all three sides 1 π/2π

16 Graph y = (π/2)+Arcsin x XY 0 π/4 0 π/2 3π/4 1 π Since we are graphing Arcsin the domain will become the range, but it will change!! Solve for x: Domain is now: Add π/2 to all three sides Take the sine of both sides What is the domain for Sin(x)? 1 π/2 π Now make a table using the y-values as your input into this function: y = Arcsinx Just shifted up π/2

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

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

19 Graph:

20 Determine if each of the following is true or false. If false give a counter example. 1. Cos -1 (cos x) = x for all values of x 1. Sin -1 (sin x) = x for all values of x 2. Sin -1 x + Cos -1 x = π /2 -1 ≤ x ≤ 1 3. Arccos x = Arccos (-x) -1 ≤ x ≤ 1 4. Tan -1 x = 1/(Tan x) x = 270° Cos -1 (cos 270°) = Cos -1 (0) = 90° FALSE TRUE FALSE x = 180° or try x = 225° Sin -1 (sin 180°) = Sin -1 (0) = 0° x = 0 of try x = 1 or -1 Sin -1 (0) + Cos -1 (0) = 0° + 90°= 90° x = 1 or try x = -1 Arccos(1) ≠ Arccos (-1) 0° ≠ 180° x = 0 Tan -1 (0) ≠ 1 / (Tan (0)) 0° ≠ 1 / 0 UNDEFINDED

21 Evaluate each expression -30 degrees 45 degrees

22 Evaluate each expression 1 Negative square root of 3


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