# How do you determine the quadrant in which 6.02 radians lies?

Aug 18, 2017

I tried this:

#### Explanation:

Radians are a bit difficult and confusing because it is not easy to "see" the angle as in degrees.
We know that an angle of ${360}^{\circ}$ corresponds to $2 \pi = 6.28$ radians so that $6.02$ radians should be a bit less than ${360}^{\circ}$ and lie in the fourth quadrant.

When I have an angle in radians and I want to place it in terms of quadrants I always try to evaluate mentally the position by referring to $\pi = 3.14$ radians knowing that if my unknown angle is less will fall either into the second or first quadrant, while if it is bigger in the third or fourth.
1st quadrant: from $0$ to $\frac{\pi}{2} = 1.57$ radians;
2nd quadrant: from $1.57$ to $\pi = 3.14$ radians:
3rd quadrant: from $3.14$ to $\frac{3}{2} \pi = 4.71$ radians:
4th quadrant: from $4.71$ to $2 \pi = 6.28$ radians.

The real trick, anyway, is experience and to use radians extensively to try to get used to them.

Aug 18, 2017

You can convert it to degrees, since it then becomes easier to figure it out.

#### Explanation:

To convert something from radians to degrees, you have to multiply it by a conversion factor of 180^@ / (pi " radians".

The logic behind this is that since ${180}^{\circ}$ and $\pi \text{ radians}$ are equal,
180^@ / (pi " radian" will be equal to $1$.

And you may know, that multiplying anything by $1$ results in the same value.

Also, more importantly, multiplying a $\text{radian}$ value by this conversion factor results in $\text{radians}$ being cancelled out from the numerator and denominator, so that you get your answer in degrees.

For this particular question, we have been given $6.02 \text{ radians}$.
Using the conversion factor mentioned above,

$6.02 \cancel{\text{radians") xx 180^@ / (3.14 cancel("radians}}$

$= 6.02 \times {180}^{\circ} / 3.14 = {345.06}^{\circ}$

Now, to locate ${345.06}^{\circ}$ on a Cartesian plane.
(Although you can guess that it will be in the 4th quadrant because it is slightly less than ${360}^{\circ}$)

In this image, we can see that ${345.06}^{\circ}$ will lie in QIV, or the 4th quadrant.

Hope this helps :)

Aug 18, 2017

#### Explanation:

We know that their are $2 \pi$ radians in ${360}^{o}$.

My preferred approach is to just learn (or calculate) the quadrants, based on the $\pi$ multiple equivalent, rather than the decimal equivalent.

If we have an angle, $\theta$, outside the range $0 \le \theta < 2 \pi$ then subtract multiples of $2 \pi$ until $0 \le \theta < 2 \pi$.

Then we use the quadrant and trace map out multiples of $\frac{\pi}{2}$ anti-clockwise:

So if we take the angle $\theta = {6.02}^{c}$ and divide by $\pi$ we get $1.91 \ldots$, and as $\frac{3}{2} < 1.91 < 2$, then $\theta = {6.02}^{c}$ lies in Quadrant IV