How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i}$ $e^( ( pi)/4 i) - e^( ( 7 pi)/4 i)$ using trigonometric functions?

Question

How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i}$ $e^( ( pi)/4 i) - e^( ( 7 pi)/4 i)$ using trigonometric functions?

1 Answer

Impact of this question

1440 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-27T09:23:17

$e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i} = \sqrt{2} i$ $e^((pi)/4i)-e^((7pi)/4i)=sqrt2i$

Explanation:

As $e^{i θ} = cos θ + i sin θ$ $e^(itheta)=costheta+isintheta$ , we have

$e^{\frac{π}{4} i} = cos (\frac{π}{4}) + i sin (\frac{π}{4})$ $e^((pi)/4i)=cos((pi)/4)+isin((pi)/4)$ and

$e^{\frac{7 π}{4} i} = cos (\frac{7 π}{4}) + i sin (\frac{7 π}{4})$ $e^((7pi)/4i)=cos((7pi)/4)+isin((7pi)/4)$

Hence, $e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i}$ $e^((pi)/4i)-e^((7pi)/4i)$

= $(cos (\frac{π}{4}) + i sin (\frac{π}{4})) - (cos (\frac{7 π}{4}) + i sin (\frac{7 π}{4}))$ $(cos((pi)/4)+isin((pi)/4))-(cos((7pi)/4)+isin((7pi)/4))$

As $cos (\frac{π}{4}) = cos (\frac{7 π}{4}) = \frac{1}{\sqrt{2}}$ $cos(pi/4)=cos((7pi)/4)=1/sqrt2$ ,

$sin (\frac{π}{4}) = \frac{1}{\sqrt{2}}$ $sin(pi/4)=1/sqrt2$ and $sin (\frac{7 π}{4}) = - \frac{1}{\sqrt{2}}$ $sin((7pi)/4)=-1/sqrt2$

$e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i}$ $e^((pi)/4i)-e^((7pi)/4i)$

= $(\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}) - (\frac{1}{\sqrt{2}} + i (- \frac{1}{\sqrt{2}}))$ $(1/sqrt2+i1/sqrt2)-(1/sqrt2+i(-1/sqrt2))$

= $(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} + i (\frac{1}{\sqrt{2}}) + i (\frac{1}{\sqrt{2}}))$ $(1/sqrt2-1/sqrt2+i(1/sqrt2)+i(1/sqrt2))$

= $0 + i \frac{2}{\sqrt{2}}$ $0+i2/sqrt2$ = $\sqrt{2} i$ $sqrt2i$

Science

Math

Humanities

... and beyond

How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i}$ $e^( ( pi)/4 i) - e^( ( 7 pi)/4 i)$ using trigonometric functions?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you evaluate eπ4i−e7π4i e^( ( pi)/4 i) - e^( ( 7 pi)/4 i) using trigonometric functions?

1 Answer

Explanation:

Related questions

Impact of this question

How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{7 π}{4} i}$ $e^( ( pi)/4 i) - e^( ( 7 pi)/4 i)$ using trigonometric functions?