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Unformatted text preview: 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 . Detailed Solution: y = arcsin( ax ) means that sin( y ) = ax . Differentiate both sides: cos( y ) y prime = a y prime = a/ cos( y ) = a/ radicalBig 1- sin 2 ( y ) = a/ 1- a 2 x 2 . (We have used the trigonometric identity sin 2 ( x ) + cos 2 ( x ) = 1 to express the result in terms of the sine function, and hence in terms of x .) 11.2 The inverse trigonometric function arctan( x ) (also written arctan( x )) means the angle where- / 2 < < / 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 ). v.2005.1 - September 4, 2009 1 Math 102 Problems Chapter 11 Detailed Solution: (a) arcsin( x ) is the angle, whose sine is x . Lets call it . We can draw a right triangle, with angle , opposite side of length x and hypotenuse 1 to get this relationship. Thus, sin( ) = sin(arcsin( x )) = x/ 1 = x....
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- Winter '09