Question

Write the polar form of a complex number in standard form for D%7B6%7D%29%20%2B%20isin%28%5Cfrac%7B5%5Cpi%20%7D%7B6%7D%29%5D" id="TexFormula1" title="25[cos(\frac{5\pi }{6}) + isin(\frac{5\pi }{6})]" alt="25[cos(\frac{5\pi }{6}) + isin(\frac{5\pi }{6})]" align="absmiddle" class="latex-formula">

Answer 1

Answer:

Standard Complex Form  : $-\frac{25\sqrt{3}}{2}+\frac{25}{2}i$

Step-by-step explanation:

We want to rewrite this expression in standard complex form. Let's first evaluate cos(5π / 6). Remember that cos(x) = sin(π / 2 - x). Therefore,

cos(5π / 6) = sin(π / 2 - 5π / 6),

π / 2 - 5π / 6 = - π / 3,

sin(- π / 3) = - sin(π / 3)

And we also know that sin(π / 3) = √3 / 2. So - sin(π / 3) = - √3 / 2 = cos(5π / 6).

Now let's evaluate the sin(5π / 6). Similar to cos(x) = sin(π / 2 - x), sin(x) = cos(π / 2 - x). So, sin(5π / 6) = cos(- π / 3). Now let's further simplify from here,

cos(- π / 3) = cos(π / 3)

We know that cos(π / 3) = 1 / 2. So, sin(5π / 6) = 1 / 2

Through substitution we receive the expression 25( - √3 / 2 + i(1 / 2) ). Further simplification results in the following expression. As you can see your solution is option a.

$-\frac{25\sqrt{3}}{2}+\frac{25}{2}i$