In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6 (cos 2π/3 + i sin 2π/3)

검증된 단계별 안내

Recognize that the given complex number is in polar (trigonometric) form: \(r(\cos \theta + i \sin \theta)\), where \(r = 6\) and \(\theta = \frac{2\pi}{3}\).

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: \(x = 6 \times \cos \left( \frac{2\pi}{3} \right)\).

Calculate the imaginary part: \(y = 6 \times \sin \left( \frac{2\pi}{3} \right)\).

Write the rectangular form as \(x + yi\), substituting the values found for \(x\) and \(y\). If necessary, round the values to the nearest tenth.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

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 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.

Evaluating Trigonometric Functions at Specific Angles

Calculating cos θ and sin θ for angles like 2π/3 involves understanding unit circle values. For 2π/3, cos 2π/3 = -1/2 and sin 2π/3 = √3/2, which are essential for accurate conversion.