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

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.