1. Inverse Trig Functions Modified from the lesson at: http://www.letu.edu/.../Lesson3.5InverseTrig Functions....%20Cached%20Similar

2. * Start with Sine Function  Given y = sin (x)

3. * Start with Sine Function  Given y = sin (x)  Table of values xy = sin(x) -3.14160.0000 -2.6180-0.5000 -2.0944-0.8660 -1.5708 -1.0472-0.8660 -0.5236-0.5000 0.0000 0.52360.5000 1.04720.8660 1.57081.0000 2.09440.8660 2.61800.5000 3.14160.0000 3.6652-0.5000 4.1888-0.8660 4.7124 5.2360-0.8660 5.7596-0.5000 6.28320.0000

4. * Start with Sine Function  Given y = sin (x)  Table of values  Graph xy = sin(x) -3.14160.0000 -2.6180-0.5000 -2.0944-0.8660 -1.5708 -1.0472-0.8660 -0.5236-0.5000 0.0000 0.52360.5000 1.04720.8660 1.57081.0000 2.09440.8660 2.61800.5000 3.14160.0000 3.6652-0.5000 4.1888-0.8660 4.7124 5.2360-0.8660 5.7596-0.5000 6.28320.0000

5. * Start with Sine Function  Given y = sin (x)  Table of values  Graph xy = sin(x) -3.14160.0000 -2.6180-0.5000 -2.0944-0.8660 -1.5708 -1.0472-0.8660 -0.5236-0.5000 0.0000 0.52360.5000 1.04720.8660 1.57081.0000 2.09440.8660 2.61800.5000 3.14160.0000 3.6652-0.5000 4.1888-0.8660 4.7124 5.2360-0.8660 5.7596-0.5000 6.28320.0000 What if we reversed the ordered pairs … y for x ?

6. * Reversed Ordered Pairs xy 0.0000-3.1416 -0.5000-2.6180 -0.8660-2.0944 -1.5708 -0.8660-1.0472 -0.5000-0.5236 0.0000 0.50000.5236 0.86601.0472 1.00001.5708 0.86602.0944 0.50002.6180 0.00003.1416 -0.50003.6652 -0.86604.1888 4.7124 -0.86605.2360 -0.50005.7596 0.00006.2832

7. * Reversed Ordered Pairs  Problem This is not a function Fails the vertical line test There are multiple (x,y)'s where x =.5 xy 0.0000-3.1416 -0.5000-2.6180 -0.8660-2.0944 -1.5708 -0.8660-1.0472 -0.5000-0.5236 0.0000 0.50000.5236 0.86601.0472 1.00001.5708 0.86602.0944 0.50002.6180 0.00003.1416 -0.50003.6652 -0.86604.1888 4.7124 -0.86605.2360 -0.50005.7596 0.00006.2832

8. * Reversed Ordered Pairs  Problem This is not a function Fails the vertical line test There are multiple (x,y)'s where x =.5  Solution Limit the range xy 0.0000-3.1416 -0.5000-2.6180 -0.8660-2.0944 -1.5708 -0.8660-1.0472 -0.5000-0.5236 0.0000 0.50000.5236 0.86601.0472 1.00001.5708 0.86602.0944 0.50002.6180 0.00003.1416 -0.50003.6652 -0.86604.1888 4.7124 -0.86605.2360 -0.50005.7596 0.00006.2832

9. * The Inverse Trig Function  We say  Similarly for inverse cosine  The range of cos -1 x is limited differently

10. * Evaluating Inverse Functions  Consider cos -1 (-0.5) We are asking what angle has a cosine value of -0.5

11. * Evaluating Inverse Functions  Consider cos -1 (-0.5) We are asking what angle has a cosine value of -0.5  Cosine negative in quadrants 2 and 3 But for cos -1 (x) we look only in 1 & 2

12. * Evaluating Inverse Functions  Consider cos -1 (-0.5) We are asking what angle has a cosine value of -0.5  Cosine negative in quadrants 2 and 3 But for cos -1 (x) we look only in 1 & 2 2

13. * Evaluating Inverse Functions  Consider cos -1 (-0.5) We are asking what angle has a cosine value of -0.5  Cosine negative in quadrants 2 and 3 But for cos -1 (x) we look only in 1 & 2 2 Calculator also capable of evaluating inverse trig functions

14. * Evaluating Inverse Functions  Consider cos -1 (-0.5) We are asking what angle has a cosine value of -0.5  Cosine negative in quadrants 2 and 3 But for cos -1 (x) we look only in 1 & 2 2 Calculator also capable of evaluating inverse trig functions

15. * Try It Out  Consider these

16. * Try It Out  Consider these

17. * Try It Out  Consider these

18. http://www.prenhall.com/divisions/esm/app/graphing/ti83/Home _Screen/Function_Keys/trig_keys/trig_keys.html Calculator Notes Cosecant, Secant, and Cotangent are not built in to the TI-83 or the TI-84 so you will have to use the reciprocal key or put the expression in rational form to evaluate. You have to be careful with coefficients. The coefficients should not be placed in the denominator.

19. * Try It Out  Consider these

20. Try It Out  Consider these

21. Composition of Trig Functions and Inverses  Try these …

22. Composition of Trig Functions and Inverses  Try these …

23. Composition of Trig Functions and Inverses  Try these …

24. Composition of Trig Functions and Inverses  Try these …