MATH 31A Midterm: Math 31A Exam 10
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Hint: sin2 x + cos2 x = 1; the quadratic formula is awesome!) In our case we need to have f (c) = f ( . 2 ) arctan(sin 0) arctan(1) arctan(0) Taking the derivative of the function (using the chain rule and the rules for the derivatives of the sine and arctangent functions) we have f (x) = 1 + (sin x)2 cos x = cos x. So we need f (c) = cos c. 2 or 2 cos c = 1 + sin2 c. Using sin2 c + cos2 c = 1 this can be rewritten as. 2 cos c = 2 cos2 c or cos2 c + 2 cos c 2 = 0. This is a quadratic (but with cos c instead of x) so we can use the quadratic formula to nd cos c = 2 q22 4 1 ( 2) But 1 3 < 1 and so can never be the cosine of an angle.
