Chapter 9: Q. 48 (page 592)

Write the expression in the standard form $a + b i$ .
${[\sqrt{3} (\cos \frac{5 π}{18} + i \sin \frac{5 π}{18})]}^{6}$

Short Answer

Expert verified

The expression in the standard form is $\frac{27}{2} - \frac{27 \sqrt{3}}{2} i$ .

Step by step solution

Step 1. Given information.

Consider the given question,

${[\sqrt{3} (\cos \frac{5 π}{18} + i \sin \frac{5 π}{18})]}^{6}$

We know that for a complex number raise to n when it is in the polar form $z^{n} = r^{n} (\cos θ + i \sin (n θ)) . . . . . . (i)$

Write the expression in the standard form $z = (a + i b)$ ,

$z^{6} = {[\sqrt{3} (\cos \frac{5 π}{18} + i \sin \frac{5 π}{18})]}^{6}$

Step 2. Use equation (i) and solve the equation.

Using the equation (i),

$z^{6} = {(\sqrt{3})}^{6} [\cos (6 \cdot \frac{5 π}{18}) + i \sin (6 \cdot \frac{5 π}{18})] z^{6} = 3^{3} (\cos \frac{5 π}{18} + i \sin \frac{5 π}{18}) z^{6} = 27 (\frac{1}{2} - \frac{\sqrt{3}}{2} i) z^{6} = \frac{27}{2} - \frac{27 \sqrt{3}}{2} i$

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Write the expression in the standard form $a + b i$ .
${[\sqrt{3} (\cos \frac{5 π}{18} + i \sin \frac{5 π}{18})]}^{6}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use equation (i) and solve the equation.

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use equation (i) and solve the equation.

One App. One Place for Learning.

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Write the expression in the standard form $a + b i$ .
${[\sqrt{3} (\cos \frac{5 π}{18} + i \sin \frac{5 π}{18})]}^{6}$