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

Given complex number: We are given the complex number in polar form: 3[cos((pi)/(2)) + i sin((pi)/(2))]. To convert it to rectangular form, we need to evaluate the cosine and sine functions.Evaluate trigonometric functions: Evaluate cos((pi)/(2)) and sin((pi)/(2)). We know that cos((pi)/(2)) = 0 and sin((pi)/(2)) = 1.Substitute values and simplify: Substitute the values of cos((pi)/(2)) and sin((pi)/(2)) into the expression. This gives us 3[0 + i * 1].Final rectangular form: Simplify the expression by multiplying 3 with each term inside the brackets. This results in 0 + 3i.The rectangular form of the complex number is 0 + 3i, which can be written simply as 3i.

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Precalculus

Convert complex numbers between rectangular and polar form

Full solution

Q. Write

3\left[\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right]

in rectangular form.

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Simplify any radicals.

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