Using Euler Solution cos(x + y) + i sin(x + y) = ei(x+y) = eix eiy = (cos(x) + i

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Using Euler

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cos(x + y) + i sin(x + y) = ei(x+y) = eix eiy = (cos(x) + i sin(x))(cos(y) + i sin(y)) = cos(x) cos(y) - sin(x) sin(y) + i cos(x) sin(y) + i sin(x) cos(y) by equating real and imaginary parts. The ranges of the inverse trig functions are given by: The range of arcsin is [-p/2, p/2], The range of arccos is [0, p], The range of arctan is (-p/2, p/2), The range of arcsec is [0, p/2) ? (p/2, p], The range of arccsc is [-p/2, 0) ? (0, p/2]. It is sometimes convenient to de?ne arccot(x) to have range (0, p), and sometimes convenient to de?ne it to have range (-p/2, 0) ? (0, p/2]. sin(x + p) = - sin x, cos(x + p) = - cos(x), (p ) - x = sin x, 2 (p ) sin - x = cos x. 2 cos Some inverse-trig indentities: If x > 0, then p , 2 p arcsec x + arccsc x = , 2 p arctan x + arccot x = , 2 arcsin x + arccos x = arcsec x = arccos(1/x), arccsc x = arcsin(1/x), arccot x = arctan(1/x) whenever both inverse trig functions are de?ned. (The ?rst four equations are also true for x < 0; arctan(x)+ p arccot(x) = if the range of arccot is taken to be (0, p), and arccot x = arctan(1/x) if the range of arccot 2 is taken to be (-p/2, 0) ? (0, p/2].) Some limits: lim arctan x = x?8 p , 2 p lim arctan x = - , 2 x?-8 lim arcsec x = lim arcsec x = x?-8 x?8 It’s also good to be able to compute any trig function of any inverse trig function. For example, tan(arcsec x) = v x2 - 1, ( ) 4 4 cos arccot = . 3 5

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