Answering problems

Many have heard that el" + 1=0, where i = V-1 . AP Calculus (BC) students now enough Math show why this is true. First, let's review some imaginary numbers. Fill in the blanks, simplifying final answers when possible. i =V-1 12 = _ 13 = 14 = is = 16 = _ i' = _ 18 = Now, we must "prove" the following identity: ex = cos(x) - isin(x). To do so, write the sixth-degree Maclaurin polynomial for e* . Then, use that to write the sixth-degree Maclaurin polynomial for e-ix . Now, rearrange the result. Put all terms that do not involve i in one set of parentheses; and then put all terms that do involve i in a different set of parentheses. Your answer here should have the form (sum of terms with real coefficients) + (sum of terms with imaginary coefficients) . actor out - i from every term in the second set of parentheses. Here, your answer should have the form (sum of terms) + (-i).(sum of terms). What function is modeled by the series in the first set of parentheses? What function is modeled by the series multiplying I in the second set of parentheses? Now, take the identity e " = cos(x) - isin(x), substitute - 7 for x on each side, and simplify the result on the right. Then, rearrange the equation so that all terms equal zero