In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6 (cos 2π/3 + i sin 2π/3)

Verified step by step guidance

1. Recognize that the given complex number is in polar (trigonometric) form: \(r(\cos \theta + i \sin \theta)\), where \(r = 6\) and \(\theta = \frac{2\pi}{3}\).
2. Recall that to convert from polar form to rectangular form, use the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(x\) is the real part and \(y\) is the imaginary part.
3. Calculate the real part: \(x = 6 \times \cos \left( \frac{2\pi}{3} \right)\).
4. Calculate the imaginary part: \(y = 6 \times \sin \left( \frac{2\pi}{3} \right)\).
5. Write the rectangular form as \(x + yi\), substituting the values found for \(x\) and \(y\). If necessary, round the values to the nearest tenth.

Key Concepts

Polar Form of Complex Numbers

A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form highlights the geometric interpretation of complex numbers on the complex plane.

Conversion from Polar to Rectangular Form

To convert a complex number from polar to rectangular form, use the formulas x = r cos θ and y = r sin θ, where x is the real part and y is the imaginary part. This allows expressing the number as x + iy.

Evaluating Trigonometric Functions at Specific Angles

Calculating cos θ and sin θ for angles like 2π/3 involves understanding unit circle values. For 2π/3, cos 2π/3 = -1/2 and sin 2π/3 = √3/2, which are essential for accurate conversion.