The image shows the solution to a problem involving converting a complex number into polar form. Let me break it down step by step:

1. Complex Number Representation

The complex number is given as $z = 8 + 6i$.

2. Rectangular to Polar Conversion

• $z = x + iy$, where $x = 8$ and $y = 6$.
• The modulus $r$ is calculated as: $r = \sqrt{x^2 + y^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10.$
• The argument $\theta$ is: $\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{6}{8}\right) = \tan^{-1}(0.75).$ From the image, it appears $\tan^{-1}(0.75) \approx 36.8^\circ$, which is converted to radians if needed.

3. Polar Form of the Complex Number

Using the polar form formula: $z = r \left(\cos\theta + i\sin\theta\right),$ substituting $r = 10$ and $\theta \approx 36.8^\circ$: $z = 10 \left(\cos 36.8^\circ + i \sin 36.8^\circ\right).$

This final step provides the polar representation.

Do you have any questions about this? Would you like me to provide details or check a specific part?

Related Questions:

1. How do you calculate $\tan^{-1}(x)$ for angles in radians and degrees?
2. Why is the modulus $r$ crucial in polar representation?
3. Can you convert back from polar form to rectangular form?
4. What happens when the angle $\theta$ is in a different quadrant?
5. How does Euler's formula $z = re^{i\theta}$ relate to the polar form?

Tip:

Always check if $\theta$ falls in the correct quadrant using the signs of $x$ and $y$.