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# Hprec8 2

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### Hprec8 2

1. 1. 8.2: Inverse Trig Functions © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Review Special Angles •Evaluate inverse trig functions.
2. 2. Review The special angles are: 6 π 3 π 60° or 4 π 45° or 30° or
3. 3. Consider the first two special angles in degrees... 30° 60° Long Side ShortSide Hypotenuse Review
4. 4. And in radians: Long Side ShortSide Hypotenuse Review 6π 3π
5. 5. Important Idea In a 30° ,60° , triangle: •the short side is one-half the hypotenuse. •the long side is times the short side. 3 ( 6)π ( 3)π ( 2)π90°
6. 6. Important Idea In a 45° ,45° ,90° triangle: •The legs of the triangle are equal. •the hypotenuse is times the length of the leg. 2 ( 4)π ( 4)π ( 2)π
7. 7. Try This Find the length of the missing sides: 30° 60° 1 2 1 3 or 3
8. 8. Try This Find the length of the missing sides 1 1 2 45°
9. 9. Important Idea Many trig functions can be solved without graphing by using special angles and inverse trig functions. A special angle solution will be an exact solution whereas a graphing solution is only approximate.
10. 10. Definition Trig Function Inverse Trig Function siny x= 1 sinx y− = cosy x= 1 cosx y− = tany x= 1 tanx y− =
11. 11. Important Idea arcsin y arccos y arctan y In some books: 1 sin y− 1 cos y− 1 tan y− instead of instead of instead of
12. 12. Example Find the exact value without using a calculator; 1 1 sin 2 − 1 2 cos 2 − ( )1 sin 1− − 1 tan (1)− 1 3 cos 2 −   − ÷  
13. 13. Example Find all values of x in the interval for which: 2 cos 2 x = 0 360x° ≤ ≤ °
14. 14. Example Find all values of x in the interval for which: 2 cos 2 x = 0 2x π≤ ≤
15. 15. Try This Find all values of x in the interval for which: 1 sin 2 x = − 0 360x° ≤ ≤ ° 210 & 330x = ° °
16. 16. Try This Find all values of x in the interval for which tan 1x = 0 360x° ≤ ≤ ° 45 & 225x = ° ° Hint: in what quadrants is the tangent positive?
17. 17. Example Write each equation in the form of an inverse relation: 4 tan 5 α=
18. 18. Example Write each equation in the form of an inverse relation: 1 cos 3 β =
19. 19. Try This Write each equation in the form of an inverse relation: sin 1x = 1 sin 1 or arcsin1x x− = =
20. 20. Example Find the value of x in the for which: cos .6328x = Leave your answer in degrees to the nearest tenth.
21. 21. Try This Find the value of x in the for which: sin .6328x = Leave your answer in degrees to the nearest tenth. x=39.3 Can you find another value for x?
22. 22. Definition The calculator will provide only the Principal Values of inverse trig functions: 1 sin x− 1 cos x− 1 tan ( )x− [ ]90 ,90− ° ° [ ]2, 2π π− [ ]0,180° [ ]0,πor or
23. 23. Example Evaluate the expression. Assume all angles are in quadrant 1. 1 sin arcsin 2    ÷  
24. 24. Example Evaluate the expression. Assume all angles are in quadrant 1. 3 sin arccos 2    ÷ ÷  
25. 25. Try This Evaluate the expression. Assume all angles are in quadrant 1. 1 os arcsinc 2    ÷   3 2
26. 26. Racetrack curves are banked so that cars can make turns at high speeds. The proper banking angle,θ, is given by: 2 tan v gr θ = where v is the velocity of the car, g is the acceleration of gravity & r is the radius of the turn. Find θ when r=1000ft & v=180 mph. Example
27. 27. Lesson Close In order to have an inverse trig function, we must restrict the domain so that duplicate values are eliminated.