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Chapter 6: Problem 27

Write each complex number in rectangular form. If necessary, round to the nearest tenth. $$6\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$$

Short Answer

Expert verified
The rectangular form of \(6\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\) is \(3\sqrt{3}+3i\).

Step by step solution

01

Identify the parameters of the polar form

In this case, the modulus, \(r\), is 6 and the argument, \(\theta\), is 30 degrees.
02

Calculate the real part

The real part, \(a\), of the complex number is found using the formula \(a = r\cos\theta\). Therefore, \(a = 6\cos30 = 6*\frac{\sqrt{3}}{2} = 3\sqrt{3}\).
03

Calculate the imaginary part

The imaginary part, \(b\), of the complex number is found using the formula \(b = r\sin\theta\). Therefore, \(b = 6\sin30 = 6*\frac{1}{2} = 3\).
04

Write out in rectangular form

The rectangular form of the complex number is written as \(a+bi\). In this case, it becomes \(3\sqrt{3}+3i\).

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Most popular questions from this chapter

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$5 \mathbf{u} \cdot(3 \mathbf{v}-4 \mathbf{w})$$

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