Point P(t) on unit circle

Hi!

Show the approximate location on the unit circle of $\displaystyle P(t)$ for the given value of $\displaystyle t$.

1. $\displaystyle t = -\frac{37\pi}{6} $

2. $\displaystyle t = -\frac{4\pi}{3}$

3. $\displaystyle t = -\frac{7\pi}{4}$

**What's the 'process' behind showing where the points are?**

I have tried adding/subtracting $\displaystyle 2\pi$ to obtain a point on the unit circle so I can have a general idea. It wasn't concrete.

You can use #1. as an example and I'll do the rest and post.

Re: Point P(t) on unit circle

For the first one, I would write:

$\displaystyle -\frac{37\pi}{6}=-\frac{36\pi+\pi}{6}=-6\pi-\frac{\pi}{6}=3(-2\pi)-\frac{\pi}{6}$

Now you can see this is 3 complete revolutions in the clockwise (negative) direction plus an additional negative amount...

Re: Point P(t) on unit circle

Quote:

Originally Posted by

**MarkFL** For the first one, I would write:

$\displaystyle -\frac{37\pi}{6}=-\frac{36\pi+\pi}{6}=-6\pi-\frac{\pi}{6}=3(-2\pi)-\frac{\pi}{6}$

Now you can see this is 3 complete revolutions in the clockwise (negative) direction plus an additional negative amount...

But how do you know where to start, in order to go the 3 revolutions?

You can't start at $\displaystyle -2\pi$ or $\displaystyle -\frac\pi6$.

Re: Point P(t) on unit circle

You start at $\displaystyle t=0$ which is the point (1,0)...

Re: Point P(t) on unit circle

$\displaystyle t = -\frac{4\pi}{3}$

$\displaystyle -\frac{4\pi - \pi}{3}$

$\displaystyle -\frac13\left(4\pi - \pi\right)$

$\displaystyle 4\left(-\frac\pi3 - \frac{\pi}{3}\right)$

So, that is 4 clockwise revolutions from $\displaystyle \frac\pi3$, which lands us back on $\displaystyle \frac\pi3$, and from there we move $\displaystyle \frac\pi3$ counterclockwise to get $\displaystyle \frac{2\pi}{3}$ on the unit circle which is the approximate location of $\displaystyle -\frac{4\pi}{3}$

ASIDE:

Does this mean $\displaystyle -\frac{36\pi+\pi}{6}= -\left(\frac{36\pi+\pi}{6}\right)$?

I was unsure because it led to: $\displaystyle \-6\pi-\frac{\pi}{6}$. It appears you factored.

Re: Point P(t) on unit circle

Quote:

Originally Posted by

**Unreal** $\displaystyle t = -\frac{4\pi}{3}$

$\displaystyle -\frac{4\pi - \pi}{3}$

$\displaystyle -\frac13\left(4\pi - \pi\right)$

$\displaystyle 4\left(-\frac\pi3 - \frac{\pi}{3}\right)$

So, that is 4 clockwise revolutions from $\displaystyle \frac\pi3$, which lands us back on $\displaystyle \frac\pi3$, and from there we move $\displaystyle \frac\pi3$ counterclockwise to get $\displaystyle \frac{2\pi}{3}$ on the unit circle which is the approximate location of $\displaystyle -\frac{4\pi}{3}$

No, this is incorrect...$\displaystyle -\frac{4\pi}{3}$ is less than one complete revolution, so you simply move $\displaystyle \frac{4\pi}{3}$ radians in the negative (clockwise) direction. Since:

$\displaystyle \frac{\frac{4\pi}{3}}{2\pi}=\frac{2}{3}$, we know we need to make 2/3 of a complete revolution.

Quote:

Originally Posted by

**Unreal** ASIDE:

Does this mean $\displaystyle -\frac{36\pi+\pi}{6}= -\left(\frac{36\pi+\pi}{6}\right)$?

I was unsure because it led to: $\displaystyle -6\pi-\frac{\pi}{6}$. It appears you factored.

Yes, both are the same...$\displaystyle -\frac{a+b}{c}=-\left(\frac{a+b}{c} \right)$.

Re: Point P(t) on unit circle

Quote:

Originally Posted by **MarkFL**

Since:

$\displaystyle \frac{\frac{4\pi}{3}}{2\pi}=\frac{2}{3}$, we know we need to make 2/3 of a complete revolution.

How do you know where 2/3 lands?

It's a bit of an awkward number.

$\displaystyle t = -\frac{7\pi}{4}$

$\displaystyle \frac{\frac{7\pi}{4}}{2\pi} = \frac78$

So, we are moving 7/8 of a revolution clockwise, which lands on $\displaystyle \frac\pi4$.

The circle is divided into 4 parts, so in this case starting from (1,0) and counting clockwise, the third part is 75%. I then used this to approximate where 7/8 is the denominator is 4, so $\displaystyle \frac\pi4$

Is $\displaystyle t = 0$ always the starting point?

Re: Point P(t) on unit circle

Quote:

Originally Posted by

**Unreal** How do you know where 2/3 lands?

It's a bit of an awkward number.

It may be easier to just think of it in its original form...$\displaystyle -\frac{4\pi}{3}=-\pi-\frac{\pi}{3}$

So go half way around in the negative direction, then continue a third of another half...

Quote:

Originally Posted by

**Unreal** $\displaystyle t = -\frac{7\pi}{4}$

$\displaystyle \frac{\frac{7\pi}{4}}{2\pi} = \frac78$

So, we are moving 7/8 of a revolution clockwise, which lands on $\displaystyle \frac\pi4$.

The circle is divided into 4 parts, so in this case starting from (1,0) and counting clockwise, the third part is 75%. I then used this to approximate where 7/8 is the denominator is 4, so $\displaystyle \frac{\pi}{4}$

You can always add an integral multiple of $\displaystyle 2\pi$ to get to the same place...:

$\displaystyle -\frac{7\pi}{4}+2\pi=\frac{\pi}{4}$

Quote:

Originally Posted by

**Unreal** Is $\displaystyle t = 0$ always the starting point?

Yes, that's where we begin when we measure an angle on the unit circle.

Re: Point P(t) on unit circle

With due regards I have the following suggestion:

Whatsoever is

What so ever is the value of the angle write it in proper fraction e.g.,

23/4 π=(5+ 3/4 )π=4π+ π+ 3/4 π= π+ 3/4 π [ because even multiples ( positive or negative ) of π we reach OX the starting ray.

Now suppose it was negative we write it as sum of positive multiples of π and then subtract the angle. For example:

- 23/4 π=(-6+ 1/4 )π=-6π+ 1/4 π= 1/4 π [ because even multiples ( positive or negative ) of π we reach OX the starting ray.