We begin by recalling that polar coordinates are expressed in the form \(\displaystyle \small (r,\theta)\), where \(\displaystyle \small r\) is the radius (the distance from the origin to the point) and \(\displaystyle \small \theta\) is the angle formed between the postive x-axis and the radius.

We can find our x-coordinate and y-coordinate in rectangular form quite easily by keeping in mind two equations.

\(\displaystyle \small cos(\theta)=\frac{x}{r}\)       or          \(\displaystyle \small x=r(cos(\theta))\)

\(\displaystyle \small sin(\theta)=\frac{y}{r}\)       or          \(\displaystyle \small y=r(sin(\theta))\)

Substituting in both of these gives respectively

\(\displaystyle \small x=(10)cos(150^{\circ})=10(-\frac{\sqrt3}{2})=-5\sqrt3\)

\(\displaystyle \small y=10(sin(150^\circ))=10(\frac{1}{2})=5\)

Therefore, the rectangular coordinates of our point are \(\displaystyle \small (-5\sqrt3,5)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=5\cos60=2.5\)

\(\displaystyle y=5sin60=4.33\)

The rectangular coordinates are \(\displaystyle (2.5, 4.33)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=8\cos 45=5.66\)

\(\displaystyle y=8 \sin 45=5.66\)

The rectangular coordinates are \(\displaystyle (5.66, 5.66)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=10\cos 35=8.19\)

\(\displaystyle y=10\sin 35=5.74\)

The rectangular coordinates are \(\displaystyle (8.19, 5.74)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=4\cos \pi=4(-1)=-4\)

\(\displaystyle y=4 \sin \pi=4(0)=0\)

The rectangular coordinates are \(\displaystyle (-4, 0)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=5\cos \frac{2\pi}{3}=5\left(-\frac{1}{2}\right)=-\frac{5}{2}\)

\(\displaystyle y=5 \sin \frac{2\pi}{3}=5\left(\frac{\sqrt3}{2}\right)=\frac{5\sqrt3}{2}\)

The rectangular coordinates are \(\displaystyle \left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=10\cos \frac{4\pi}{3}=10\left(-\frac{1}{2}\right)=-5\)

\(\displaystyle y=10 \sin \frac{4\pi}{3}=10\left(-\frac{\sqrt3}{2}\right)=-5\sqrt3\)

The rectangular coordinates are \(\displaystyle (-5, -5\sqrt3)\)

To convert polar coordinates \(\displaystyle (r, \theta)\) to rectangular coordinates \(\displaystyle (x, y)\),

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

Using the information given in the question, 

\(\displaystyle x=8\cos \frac{7\pi}{6}=8\left(-\frac{\sqrt3}{2}\right)=-\frac{8\sqrt3}{2}\)

\(\displaystyle y=8 \sin \frac{7\pi}{6}=8\left(-\frac{1}{2}\right)=-4\)

The rectangular coordinates are \(\displaystyle \left(-\frac{8\sqrt3}{2}, -4\right)\)

The polar coordinates given have an angle of \(\displaystyle \small \frac{\pi}{3}\) but a negative radius, so our coordinates are located in quadrant III.

This means x and y are both negative. You can figure out these x and y coordinates using trigonometric ratios, or since the angle is \(\displaystyle \small \frac{\pi}{3}\), special right triangles. 

The hypotenuse of this triangle is 5, but in the special right triangle it's 2, so we know we're multiplying each side by \(\displaystyle \small \frac{5}{2}\).

That makes the x-coordinate or adjacent side be

\(\displaystyle \small \frac{5}{2}*1 = \frac{5}{2}\)

and the y-coordinate or opposite side be

\(\displaystyle \small \frac{5}{2}*\sqrt{3} = \frac{5\sqrt3}{2}\).

In this case, once again, both are negative, so our answer is

\(\displaystyle \left (-\frac{5}{2}, -\frac{5\sqrt3}{2}\right)\).

In order to determine the rectangular coordinates, look at the triangle representing the polar coordinates:

We can see that both x and y are positive. We can figure out the x-coordinate by using the cosine:

\(\displaystyle cos\left(\frac{\pi}{10}\right) = \frac{x}{10}\)

\(\displaystyle 0.951 \approx \frac{x}{10}\) multiply both sides by 10.

\(\displaystyle 9.51 = x\)

We can figure out the y-coordinate by using the sine:

\(\displaystyle sin\left(\frac{\pi}{10}\right) = \frac{y}{10}\)

\(\displaystyle 0.309 \approx \frac{y}{10}\)

\(\displaystyle 3.09 = y\)