What is the imaginary part of the complex number  z=2+3i  after rotating it counterclockwise by  5π6  radians in the complex plane?

To find the imaginary part of the complex number \( z = 2 + 3i \) after rotating it counterclockwise by \( \frac{5\pi}{6} \) radians in the complex plane, we need to use the formula for rotating a complex number:

\[
z_{\text{rotated}} = z \cdot e^{i\theta}
\]

where \( z = 2 + 3i \) and \( \theta = \frac{5\pi}{6} \).

### Step 1: Convert to Polar Form
First, let's express \( z = 2 + 3i \) in polar form \( r \cdot e^{i\phi} \), where:

\[
r = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]

The argument \( \phi \) of \( z \) is given by:

\[
\phi = \arctan\left(\frac{3}{2}\right)
\]

Thus, in polar form, \( z = \sqrt{13} \cdot e^{i\phi} \).

### Step 2: Apply the Rotation
When we rotate \( z \) by \( \frac{5\pi}{6} \), the new argument becomes:

\[
\phi + \frac{5\pi}{6}
\]

The new complex number after rotation is:

\[
z_{\text{rotated}} = \sqrt{13} \cdot e^{i\left(\phi + \frac{5\pi}{6}\right)}
\]

### Step 3: Convert Back to Rectangular Form
Now, we convert \( z_{\text{rotated}} \) back to rectangular form:

\[
z_{\text{rotated}} = r \cdot (\cos(\phi + \frac{5\pi}{6}) + i \sin(\phi + \frac{5\pi}{6}))
\]

Here, \( r = \sqrt{13} \). The imaginary part is given by:

\[
\text{Imaginary part} = \sqrt{13} \cdot \sin\left(\arctan\left(\frac{3}{2}\right) + \frac{5\pi}{6}\right)
\]

### Step 4: Compute the Imaginary Part
Let's compute this step-by-step:

\[
\phi = \arctan\left(\frac{3}{2}\right) \approx 0.9828 \, \text{radians}
\]

\[
\phi + \frac{5\pi}{6} \approx 0.9828 + \frac{5\pi}{6} \approx 3.5834 \, \text{radians}
\]

Now, find the sine value:

\[
\sin(3.5834) \approx -0.4258
\]

Thus, the imaginary part is:

\[
\text{Imaginary part} = \sqrt{13} \times (-0.4258) \approx -1.536
\]

### Conclusion
The imaginary part of the complex number \( z = 2 + 3i \) after rotating it counterclockwise by \( \frac{5\pi}{6} \) radians in the complex plane is approximately:

\[
\boxed{-1.536}
\]

Would you like more details on any step or have any questions?

Here are 5 related questions you might be interested in:
1. How do you convert complex numbers from rectangular to polar form?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you find the modulus and argument of a complex number?
4. How does rotating a complex number relate to Euler's formula?
5. How can you perform complex number operations using matrices?

**Tip:** Understanding how to represent complex numbers in different forms (rectangular, polar) is crucial for performing complex operations such as rotations and multiplications.

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

Learn how to find the imaginary part of the complex number z = 2 + 3i after a counterclockwise rotation by 5π/6 radians. This problem covers the use of Euler's formula, converting complex numbers between polar and rectangular forms, and applying trigonometric rotations.

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