Step by step solution to "By using euler formula,one can obtain:2sin⁡(π180)=((−1)1180)89−((−1)1180)91"

WesDiectstemiwxg

Answered question

2022-03-19

By using euler formula,one can obtain:

2 \sin (\frac{π}{180}) = {({(- 1)}^{\frac{1}{180}})}^{89} - {({(- 1)}^{\frac{1}{180}})}^{91}

blindhaedzgs

Beginner2022-03-20Added 2 answers

Let's try the following approach:

- 1 = e^{π i + 2 k π i} = e^{π i (2 k + 1)}, k \in Z \Rightarrow {(- 1)}^{\frac{1}{180}} = e^{\frac{π t}{180} (2 k + 1)}

and from here

{({(- 1)}^{\frac{1}{180}})}^{89} - {({(- 1)}^{\frac{1}{180}})}^{91} = {(e^{\frac{π t}{180} (2 k + 1)})}^{89} - {(e^{\frac{π t}{180} (2 k + 1)})}^{91}

= e^{\frac{89 π t}{180} (2 k + 1)} (1 - e^{\frac{π t}{90} (2 k + 1)})

Taking the real part of the above we get

\cos \frac{89 π (2 k + 1)}{180} (1 - \cos \frac{π (2 k + 1)}{90}) + \sin \frac{89 π (2 k + 1)}{180} \sin \frac{π (2 k + 1)}{90}

= \cos \frac{89 π (2 k + 1)}{180} - \cos \frac{91 π (2 k + 1)}{180} = 2 \sin π (2 k + 1) \cdot \sin \frac{2 π (2 k + 1)}{90} = 0

The equality one before the last above follows from the trigonometric identity

\cos x - \cos y = - 2 \sin \frac{x + y}{2} \sin \frac{x - y}{2}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?