Convert the complex number from polar form to rectangular form, $a+b i$. Give an exact answer.
\[
10\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)
\]
The rectangular form is

Step 1 ：Given the complex number in polar form is \(10\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)\).

Step 2 ：The rectangular form of a complex number in polar form is given by \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the angle.

Step 3 ：In this case, \(r = 10\) and \(\theta = \frac{5\pi}{6}\).

Step 4 ：We can use the trigonometric identities \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\) and \(\sin \frac{5\pi}{6} = \frac{1}{2}\) to convert the polar form to rectangular form.

Step 5 ：Calculate the real part \(a = r \cos \theta = 10 \times -\frac{\sqrt{3}}{2} = -5\sqrt{3}\).

Step 6 ：Calculate the imaginary part \(b = r \sin \theta = 10 \times \frac{1}{2} = 5\).

Step 7 ：Final Answer: The rectangular form of the complex number is \(\boxed{-5\sqrt{3} + 5i}\).