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^{(i11\pi)/(6)}}{3√(3)\cdot e^{(i\pi)/(4)}}`

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

`(9)/(3√(3))\cdot\frac{e^{(i11\pi)/(6)}}{e^{(i\pi)/(4)}}`

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

`(9)/(3√(3))\cdot√(3)=(9√(3))/(3\cdot3)=(9√(3))/(9)=√(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^{(i11\pi)/(6)}}{e^{(i\pi)/(4)}}=e^{i((11\pi)/(6)-(\pi)/(4))}=e^{i((22\pi-3\pi)/(12))}=e^{i((19\pi)/(12))}=\cos ((19\pi)/(12))+i\sin ((19\pi)/(12))=0.2588-i0.9659`

So the final expression is:

`√(3)\cdot(0.2588-i0.9659)=0.4483-i1.673`