ET 4.7: Evaluate

1. ET 4.7: Evaluate Evaluate the inverse functions “OEO”

2. Inverse functions = Angle Measures

3. We are about to graph the inverse of some trig functions. Domain & Range switched X & Y switched What do we remember about inverse functions? Function must be one-to-one to have an inverse Graph of an inverse function is a reflection of f(x) over the line y=x

4. y = cos(x) Domain: R Range: [-1, 1] Summary: Can find arc cos of values between -1 and 1 and you will have answers to arc cos that are angle measures between 0 and Pi. Effects Range: Let’s see how. Why does cosine only have an inverse from 0 to Pi? [0, π] Function must be one-to-one. y = cos-1(x) Domain: [-1, 1] Range: R

5. Evaluate Inverse Trig Function “OEO” -1< .707 < 1 Arc cos will have an answer. [0<1.57< π]

6. If possible, find the exact value Yes, because .5 is between -1 & 1 TRY ON If -1 < x < 1 & Then cos(arccos(x)) = x and arccos(cosy) = y

7. If possible, find the exact value x Error, because Pi isn’t between -1 and 1 TRY ON If -1 < x < 1 & Then cos(arccos(x)) = x and arccos(cosy) = y

8. If possible, find the exact value Yes, (OEO) Because inverse cos gives us an angle between -1.57 and 1.57 TRY ON If -1 < x < 1 & Then cos(arccos(x)) = x and arccos(cosy) = y

9. Graph y = arc tan (x) Summary: Can find arc tan of any value, however you will have answers to arc tan that are angle measures between -π/2 & π/2 Restrict the domain so tan is one-to-one. π/2 - π/2 π/2 - π/2 y = tan (x) R Domain: - π/2 < x < π/2 Domain: R Range: Range: - π/2 < x < π/2

10. If possible, find the exact value tan[arctan (-5) ] = -5 TRY ON MATCH! Because -5 lies in the domain of the arctan function, the inverse property applies. If x is R & Then tan(arctan(x)) = x and arctan(tany) = y

11. Graph y = arc sin (x) Summary: Can find arc sin of values between -1 and 1 and you will have answers to arc sin that are angle measures between -π/2 and π/2. Restrict the domain so sin is one-to-one. -π/2 -π/2 y = sin (x) Domain: -π/2 < x < π/2 Domain: -1 < x < 1 Range: -1 < x < 1 Range: -π/2 < x < π/2

12. If possible, find the exact value x If -1 < x < 1 & Then sin(arcsin(x)) = x and arcsin(siny) = y TRY ON MATCH!

13. When giving an answer you must consider location and value! Location: OEO Sam I am cOssEccOtcScsIntAn Value: 0<answer<3.14 -1.57<x<1.57 Inverse Properties of Trig Functions If -1 < x < 1 & Then sin(arcsin(x)) = x and arcsin(siny) = y If -1 < x < 1 & Then cos(arccos(x)) = x and arccos(cosy) = y If x is R & Then tan(arctan(x)) = x and arctan(tany) = y

14. Combine the use of your calculator, the inverse trig properties, and knowledge of OEO/Sam I am and you’ll be just fine. It is not enough to *Blindly use property arcfunc(func(x)) = x func(arc func(x)) = x * To enter in your calculator & look for a match. But, This will tell you if there is an answer or not. You just have to use OEO to check to see if you need to use a coterminal angle instead.

15. 4.7 Assignments • Day 1: 1-16 all, 19-35, 43-48, 75, 77

16. Use an inverse trig function to write theta as a function of x. Use a trig function to write an equation containing x and theta. ET 4.7b Opposite Adjacent

17. Evaluate Composition Function Create a triangle with this info combined with “OEO” 3 θ 2

18. Evaluate Composition Function Create a triangle with this info combined with “OEO” θ -3 5

19. Write as an algebraic expression in x. Quadrant I 1 θ 3x

20. 4.7 Assignments Day 2: 37-41 odd, 49-57 odd, 59-67 odd, 91, 97