Question

Divide(cos(***)+i sin(**))+Enter your answer by filling in the boxes Enter the all values as exact values in simplest form

Answer 1

To solve this question we need to use the Euler's formula, to simplify the sine and cosine expressions:

$e^{ix}=\cos (x)+i\cdot\sin (x)$

Using this formula, we can simplify the expression in the question to:

$\frac{9\cdot e^{\frac{i11\pi}{6}}}{3\sqrt{3}\cdot e^{\frac{i\pi}{4}}}$

To calculate it easier, we can separate it in a product of two fractions:

$\frac{9}{3\sqrt{3}}\cdot\frac{e^{\frac{i11\pi}{6}}}{e^{\frac{i\pi}{4}}}$

To simplify the first fraction we can rationalize it, multiplying the numerator and denominator by √3:

$\frac{9}{3\sqrt{3}}\cdot\sqrt{3}=\frac{9\sqrt{3}}{3\cdot3}=\frac{9\sqrt{3}}{9}=\sqrt{3}$

Now, to calculate the second fraction, we can use the property of dividing two exponencial numbers with the same base (we just need to subtract their exponents):

$\frac{e^{\frac{i11\pi}{6}}}{e^{\frac{i\pi}{4}}}=e^{i(\frac{11\pi}{6}-\frac{\pi}{4})}=e^{i(\frac{22\pi-3\pi}{12})}=e^{i(\frac{19\pi}{12})}=\cos (\frac{19\pi}{12})+i\sin (\frac{19\pi}{12})=0.2588-i0.9659$

So the final expression is:

$\sqrt{3}\cdot(0.2588-i0.9659)=0.4483-i1.673$