Topic 3.13 Notes – Trigonometry and Polar Coordinates

What Polar Coordinates Are

A polar point is written as $(r,\theta)$, not $(x,y)$.

• The pole is the origin.
• The polar axis is the positive $x$-axis.
• $\theta$ is the angle in standard position.
• $r$ is the signed radial displacement.

That word signed matters.

• If $r>0$, move $|r|$ units along the terminal ray of $\theta$.
• If $r<0$, move $|r|$ units in the opposite direction along the same line.
• The actual distance from the origin is $|r|$.

So $\left(-4,\frac{\pi}{6}\right)$ does not sit in Quadrant I. The angle points there, but the negative radius flips the point across the origin, as the diagram shows.

One more special case. $(0,\theta)$ is always the pole, no matter what angle you use. That leads right into the biggest difference from rectangular coordinates.

Equivalent Polar Representations

A rectangular point has one name. A polar point has infinitely many.

You get equivalent names in two ways:

$$(r,\theta)=(r,\theta+2\pi k)$$

Same radius, coterminal angles.

$$(r,\theta)=(-r,\theta+(2k+1)\pi)$$

Changing the sign of $r$ means the angle must shift by an odd multiple of $\pi$.

In degrees, those become adding $360^\circ k$ or an odd multiple of $180^\circ$.

For one point, all of these are equivalent:

A lot of quiz mistakes happen because students forget that negative $r$ flips the point across the origin.

If a problem asks for a specific form, check the instructions carefully.

• positive $r$
• angle in an interval like $0\le \theta<2\pi$
• negative angles allowed or not

Converting Between Polar and Rectangular Coordinates

Polar to rectangular

Use

$$x=r\cos\theta \qquad y=r\sin\theta$$

Example:

$$\left(-4,\frac{\pi}{6}\right)$$

$$x=-4\cos\frac{\pi}{6}=-4\cdot\frac{\sqrt3}{2}=-2\sqrt3$$

$$y=-4\sin\frac{\pi}{6}=-4\cdot\frac12=-2$$

So the rectangular coordinates are $\boxed{(-2\sqrt3,-2)}$.

Use unit-circle values when you can, and keep exact values unless told to round.

Rectangular to polar

Here’s the full move.

1. Find the radius

$$r=\sqrt{x^2+y^2}$$

2. Find the angle from

$$\tan\theta=\frac{y}{x}$$

3. Fix the quadrant

• if $x>0$, use $\theta=\arctan(y/x)$
• if $x<0$, use $\theta=\arctan(y/x)+\pi$
• if $x=0$, use $\frac{\pi}{2}$ or $\frac{3\pi}{2}$
• if $(x,y)=(0,0)$, then $r=0$ and any angle works

Example with $( -\sqrt3,1)$:

$$r=\sqrt{(-\sqrt3)^2+1^2}=\sqrt{3+1}=2$$

$$\arctan\left(\frac{1}{-\sqrt3}\right)=-\frac{\pi}{6}$$

Since $x<0$, add $\pi$:

$$\theta=-\frac{\pi}{6}+\pi=\frac{5\pi}{6}$$

So one answer is $\boxed{\left(2,\frac{5\pi}{6}\right)}$.

Another equivalent answer is $\boxed{\left(-2,-\frac{\pi}{6}\right)}$.

These formulas fit together because $x=r\cos\theta$ and $y=r\sin\theta$ give $x^2+y^2=r^2$, and dividing gives $\tan\theta=\frac{y}{x}$.

Complex Numbers in Rectangular and Polar Form

A complex number $a+bi$ matches the point $(a,b)$ in the complex plane.

• horizontal axis = real axis
• vertical axis = imaginary axis

So rectangular and polar forms describe the same number in two ways:

• Rectangular form: $z=a+bi$
• Polar form: $z=r(\cos\theta+i\sin\theta)$

The connections are

$$a=r\cos\theta,\qquad b=r\sin\theta$$

The modulus is

$$r=|z|=\sqrt{a^2+b^2}$$

The argument is the angle $\theta$. It is not unique, since $\theta+2\pi k$ gives the same direction. The complex number $0$ has no defined argument.

Examples:

• $-\sqrt3+i=2\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$
• $3\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right)=-\frac32-\frac{3\sqrt3}{2}i$

What Students Miss Under Time Pressure

• $\arctan(y/x)$ alone does not tell you the correct quadrant.
• If $x=0$, don’t use $\tan\theta=\frac{y}{x}$. Go straight to axis angles.
• A negative $r$ changes the point’s direction by $\pi$.
• Many prompts want a different equivalent polar form, not just any one answer.
• Degree mode and radian mode can wreck an otherwise correct setup.
• Early rounding can turn a clean exact answer into a messy wrong one.