3(cos 3𝜋/2+ 𝑖 sin 3𝜋/2) in standard form

To express \( 3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) \) in standard form, let's proceed step by step:

1. We are given a complex number in polar form: 
   \[
   r(\cos \theta + i \sin \theta)
   \]
   Here, \( r = 3 \) and \( \theta = \frac{3\pi}{2} \).

2. Evaluate the cosine and sine values:
   \[
   \cos \frac{3\pi}{2} = 0
   \]
   \[
   \sin \frac{3\pi}{2} = -1
   \]

3. Substitute these values back into the expression:
   \[
   3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) = 3(0 + i(-1)) = 3(0 - i) = -3i
   \]

So, the standard form of \( 3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) \) is \( -3i \).

Would you like further explanation or details? Here are some related questions to expand on this topic:

1. How do you convert from polar form to rectangular form in general?
2. What is the significance of the angle \( \frac{3\pi}{2} \) in the unit circle?
3. How do you find the modulus and argument of a complex number?
4. What is Euler's formula and how does it relate to this problem?
5. How does multiplying a complex number in polar form affect its magnitude and direction?

**Tip:** Remember that \( \cos \frac{3\pi}{2} = 0 \) and \( \sin \frac{3\pi}{2} = -1 \) correspond to the point on the negative imaginary axis in the complex plane.

3(cos 3π/2 + i sin 3π/2) in standard form

This guide shows how to convert the complex number 3(cos 3π/2 + i sin 3π/2) from polar form to rectangular form, resulting in -3i. Key concepts include polar to rectangular conversions, using trigonometric identities and Euler's formula. Suitable for Grades 10-12.

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