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

검증된 단계별 안내

Recognize that the given complex number is in polar (trigonometric) form: \(r(\cos \theta + i \sin \theta)\), where \(r = 20\) and \(\theta = 205^\circ\).

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.

Calculate the real part: \(x = 20 \times \cos 205^\circ\).

Calculate the imaginary part: \(y = 20 \times \sin 205^\circ\).

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.

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주요 개념

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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.