Write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 60° + i sin 60°)

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Recognize that the given complex number is in polar (trigonometric) form: \(r(\cos \theta + i \sin \theta)\), where \(r = 8\) and \(\theta = 60^\circ\).

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: \(x = 8 \times \cos 60^\circ\).

Calculate the imaginary part: \(y = 8 \times \sin 60^\circ\).

Write the rectangular form as \(x + yi\), substituting the values found for \(x\) and \(y\). If necessary, round the values 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 horizontal and vertical components on the complex plane.

Conversion from Polar to Rectangular Form

To convert a complex number from polar to rectangular form, multiply the magnitude r by cos θ to find the real part (a), and multiply r by sin θ to find the imaginary part (b). This yields a + bi, which is easier to interpret and use in algebraic operations.

Trigonometric Values and Rounding

Evaluating cos 60° and sin 60° requires knowledge of standard trigonometric values. After calculating the real and imaginary parts, round the results to the nearest tenth if necessary, ensuring the final rectangular form is precise and suitable for practical use.