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Write 3[cos((pi)/(2))+i sin((pi)/(2))] in rectangular form. Simplify any radicals.

Write 3[cos⁡(π2)+isin⁡(π2)] 3\left[\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right] in rectangular form.\newlineSimplify any radicals.

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Convert complex numbers between rectangular and polar formright chevron icon

Full solution

Q. Write 3[cos⁡(π2)+isin⁡(π2)] 3\left[\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right] in rectangular form.\newlineSimplify any radicals.
  1. Given complex number: We are given the complex number in polar form: 3[cos⁡(π2)+isin⁡(π2)]3[\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})]. To convert it to rectangular form, we need to evaluate the cosine and sine functions.
  2. Evaluate trigonometric functions: Evaluate cos⁡(π2)\cos\left(\frac{\pi}{2}\right) and sin⁡(π2)\sin\left(\frac{\pi}{2}\right). We know that cos⁡(π2)=0\cos\left(\frac{\pi}{2}\right) = 0 and sin⁡(π2)=1\sin\left(\frac{\pi}{2}\right) = 1.
  3. Substitute values and simplify: Substitute the values of cos⁡(π2)\cos\left(\frac{\pi}{2}\right) and sin⁡(π2)\sin\left(\frac{\pi}{2}\right) into the expression. This gives us 3[0+i⋅1]3[0 + i \cdot 1].
  4. Final rectangular form: Simplify the expression by multiplying 33 with each term inside the brackets. This results in 0+3i0 + 3i.
  5. Final rectangular form: Simplify the expression by multiplying 33 with each term inside the brackets. This results in 0+3i0 + 3i.The rectangular form of the complex number is 0+3i0 + 3i, which can be written simply as 3i3i.

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