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# Inverse Trigonometry

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### Inverse Trigonometry

1. 1. INVERSE TRIGONOMETRY
2. 2. PRINCIPAL VALUES & DOMAINS OF INVERSETRIGONOMETRIC FUNCTIONS Function Domain Range y = sin-1 x - 1 x1 y = cos-1 x - 1  x1 y = tan-1 x xR y = cosec-1 x x-1or x1 y = sec-1 x x-1or x1 y = cot-1 x xR
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4. 4. EXAMPLES
5. 5. PROPERTIES OF INVERSE TRIGONOMETRICFUNCTIONSProperty – A sin (sin-1 x) = x -1  x 1 cos (cos-1 x) = x -1  x 1 tan (tan-1 x) = x xR cot (cot-1 x) = x xR sec (sec-1 x) = x x  -1, x  1 cosec (cosec-1 x) = x x  -1, x  1These functions are equal to identity function in their wholedomain which may or may not be R.
6. 6. EXAMPLES
7. 7. Property – B sin-1 (sin x) = x cos-1 (cos x) = x tan-1 (tan x) = x cot-1 (cot x) = x sec-1 (sec x) = x cosec-1 (cosec x) = xThese are equal to identity function for a short interval of xonly.
8. 8. EXAMPLES
9. 9. Property – C sin-1 (-x) = - sin-1 x tan-1 (-x) = - tan-1 x xR xRThe functions sin-1 x, tan-1 x and cosec-1 x are oddfunctions and rest are neither even nor odd.
10. 10. EXAMPLES
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12. 12. EXAMPLES
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14. 14. EXAMPLES
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16. 16. EXAMPLES
17. 17. IDENTITIES OF ADDITION AND SUBSTRACTION x 1  y 2  y 1  x 2      x 1  y 2  y 1  x 2     
18. 18.  x y  1  x 2 1  y2      xy 1  xy xy 1  xy
19. 19. Subtraction sin-1 x - sin-1 y = sin-1 x 1  y 2  y 1  x 2  , x  0, y  0     cos-1 x - cos-1 y = cos-1 x y  1  x 1  y  ,  2 2     x  0, y  0, x  y xy tan-1 x - tan-1y = tan-1 , x  0, y  0 1  xyNote: For x < 0 and y < 0 these identities can be used withthe help of properties C i.e. change x and y to - x and - ywhich are positive.
20. 20. EXAMPLES
21. 21. SOME MORE PROPERTIES
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23. 23. EXAMPLE
24. 24. SOME PROPERTIES OF TAN-1
25. 25. SOME USEFUL GRAPHS
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