CALCULUS
hw7solutions.pdf

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

• Homework Help
• 3

This preview shows pages 2–3. Sign up to view the full content.

2

This preview has intentionally blurred sections. Sign up to view the full version.

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
This is the end of the preview. Sign up to access the rest of the document.
• Fall '12
• Hom
• Calculus, Cos, Complex number, Euler's formula, polar form, zw

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern