Chapter 9: Q. 47 (page 592)

Write the expression in the standard form $a + i b$ .
${[5 (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})]}^{4}$

Short Answer

Expert verified

The expression in the standard form is $- \frac{25 \sqrt{2}}{2} + \frac{25 \sqrt{2}}{2} i$ .

Step by step solution

Step 1. Given information.

Consider the given question,

${[5 (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})]}^{4}$

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 $z = (a + i b)$ ,

$z^{4} = {[\sqrt{5} (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})]}^{4}$

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

Using the equation (i),

$z^{4} = {(\sqrt{5})}^{4} [\cos (4 \cdot \frac{3 π}{16}) + i \sin (4 \cdot \frac{3 π}{16})] z^{4} = 5^{2} (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4}) z^{4} = 25 (- \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i) z^{4} = - \frac{25 \sqrt{2}}{2} + \frac{25 \sqrt{2}}{2} i$

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Write the expression in the standard form $a + i b$ .
${[5 (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})]}^{4}$

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.

Most popular questions from this chapter

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Write the expression in the standard form $a + i b$ .
${[5 (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})]}^{4}$