What is the imaginary part of the complex number $z=2+ 3i$ after rotating it counterclockwise by $\frac{5\pi}{6}$ radians in the complex plane?

To find the imaginary part of the complex number $ z = 2 + 3i $ after it is rotated counterclockwise by an angle of $ \frac{5\pi}{6} $ radians in the complex plane, we can use the formula for rotating a complex number.

### Step 1: Understand the Rotation Formula

When a complex number $ z = x + yi $ is rotated counterclockwise by an angle $ \theta $ in the complex plane, the new complex number $ z' $ is given by:
\[
z' = z \cdot e^{i\theta}
\]

Here, $ e^{i\theta} = \cos(\theta) + i \sin(\theta) $, using Euler's formula.

### Step 2: Calculate $ e^{i \frac{5\pi}{6}} $

Substitute $ \theta = \frac{5\pi}{6} $:

\[
e^{i \frac{5\pi}{6}} = \cos\left( \frac{5\pi}{6} \right) + i \sin\left( \frac{5\pi}{6} \right)
\]

Now, compute $ \cos\left( \frac{5\pi}{6} \right) $ and $ \sin\left( \frac{5\pi}{6} \right) $:

\[
\cos\left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}, \quad \sin\left( \frac{5\pi}{6} \right) = \frac{1}{2}
\]

Thus:

\[
e^{i \frac{5\pi}{6}} = -\frac{\sqrt{3}}{2} + i \frac{1}{2}
\]

### Step 3: Multiply the Original Complex Number by $ e^{i \frac{5\pi}{6}} $

Now, multiply $ z = 2 + 3i $ by $ e^{i \frac{5\pi}{6}} $:

\[
z' = (2 + 3i) \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right)
\]

Distribute and use the distributive property:

\[
z' = 2 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) + 3i \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right)
\]

Calculate each term:

\[
z' = \left( -\sqrt{3} + i \right) + \left( -\frac{3\sqrt{3}}{2} i + \frac{3}{2}(-1) \right)
\]

Combine like terms:

\[
z' = \left( -\sqrt{3} + \frac{3} \]








What is the imaginary part of the complex number z = 2 + 3i after rotating it counterclockwise by 5π/6 radians in the complex plane?

This problem explores how to find the imaginary part of the complex number z = 2 + 3i after rotating it counterclockwise by 5π/6 radians. The solution applies key concepts like Euler's formula and complex plane rotations, with a step-by-step breakdown.

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