2 6 write e iπ in the form a bi answer using eulers

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6. Write e - in the form a + bi . Answer: Using Euler’s formula, e - = e i ( - π ) = cos( - π ) + i sin( - π ) = - 1 + 0 i = - 1 . 7. Use Euler’s formula (i.e. e = cos θ + i sin θ ) to prove the following formulas for cos x and sin x : cos x = e ix + e - ix 2 , sin x = e ix - e - ix 2 i . Proof. Using Euler’s formula, e ix = cos x + i sin x. Similarly, e - ix = cos( - x ) + i sin( - x ), which means (using the fact that cosine is even and sine is odd) e - ix = cos x - i sin x. Therefore, e ix + e - ix 2 = (cos x + i sin x ) + (cos x - i sin x ) 2 = 2 cos x 2 = cos x, as desired. Likewise e ix - e - ix 2 i = (cos x + i sin x ) - (cos x - i sin x ) 2 i = 2 i sin x 2 i = sin x, completing the proof. 3
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