In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)

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 \(\theta = 30^\circ\).
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: \(x = 6 \times \cos 30^\circ\).
4. Calculate the imaginary part: \(y = 6 \times \sin 30^\circ\).
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 (modulus) 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.

Trigonometric Values and Rounding

Evaluating cos θ and sin θ for specific angles often requires using exact values or approximations. When the problem asks for rounding, calculate the decimal values and round to the nearest tenth to express the rectangular form accurately.