In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 20(cos 205° + i sin 205°)

Verified step by step guidance

1. Recognize that the given complex number is in polar (trigonometric) form: \(r(\cos \theta + i \sin \theta)\), where \(r = 20\) and \(\theta = 205^\circ\).
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: \(x = 20 \times \cos 205^\circ\).
4. Calculate the imaginary part: \(y = 20 \times \sin 205^\circ\).
5. Write the complex number in rectangular form as \(x + yi\), substituting the values found for \(x\) and \(y\). If necessary, round \(x\) and \(y\) to the nearest tenth.

Key Concepts

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument. The rectangular form represents the same number as a + bi, where a and b are real numbers corresponding to the x and y coordinates in the complex plane.

Conversion from Polar to Rectangular Form

To convert a complex number from polar to rectangular form, use the formulas a = r cos θ and b = r sin θ. Here, a is the real part and b is the imaginary part. This allows the expression r(cos θ + i sin θ) to be rewritten as a + bi.

Trigonometric Function Evaluation and Rounding

Evaluating cos θ and sin θ for angles like 205° requires understanding of the unit circle and possibly using a calculator. After computing these values, multiply by r and round the results to the nearest tenth if specified, ensuring the final rectangular form is accurate and concise.