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chapter11Problems

# chapter11Problems - Chapter 11 Inverse Trigonometric...

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Chapter 11 Inverse Trigonometric functions 11.1 The function y = arcsin( ax ) is a so-called inverse trigonometric function . It expresses the same relationship as does the equation ax = sin( y ). (However, this function is defined only for values of x between 1 /a and - 1 /a .) Use implicit differentiation to find y prime . 11.2 The inverse trigonometric function arctan( x ) (also written arctan( x )) means the angle θ where - π/ 2 < θ < π/ 2 whose tan is x . Thus cos(arctan( x ) (or cos(arctan( x )) is the cosine of that same angle. By using a right triangle whose sides have length 1 , x and 1 + x 2 we can verify that cos(arctan( x )) = 1 / 1 + x 2 . Use a similar geometric argument to arrive at a simplification of the following functions: (a) sin(arcsin( x )), (b) tan(arcsin( x ), (c) sin(arccos( x ). 11.3 Find the first derivative of the following functions. (a) y = arcsin x 1 3 (b) y = (arcsin x ) 1 3 (c) θ = arctan(2 r + 1) (d) y = x arcsec 1 x (e) y = x a a 2 - x 2 - arcsin x a , a > 0 . (f) y = arccos 2 t 1+ t 2 v.2005.1 - September 4, 2009 1

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Math 102 Problems Chapter 11 11.4 Your room has a window whose height is 1.5 meters.
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