First lets identify the angles that make up the third quadrant. Quadrant three is  to  or in radians,  to  thus, any angle that does not fall within this range is not in quadrant three.

Therefore, the correct answer,

is not in quadrant three because it is in the first quadrant.

This is clear when we subtract

.

First lets identify the different quadrants.

Quadrant I:

Quadrant II: 

Quadrant III: 

Quadrant IV: 

Now looking at our possible answer choices, we will add or subtract  until we get the reduced fraction of the angle. This will tell us which quadrant the angle lies in.

 thus in quadrant III. 

 thus in quadrant III.

Therefore,

 and  is the correct answer.

First lets identify the different quadrants.

Quadrant I:

Quadrant II: 

Quadrant III: 

Quadrant IV: 

The correct answer,, is coterminal with .

We can figure this out by adding , or equivalently to get , or we can count thirds of pi around the unit circle clockwise. Either way, it is the only angle that ends in the second quadrant.

One way to uncover which quadrant this angle lies is to ask how many complete revolutions this angle makes by dividing it by 360 (and rounding down to the nearest whole number).

With a calculator we find that  makes  full revolutions. Now the key lies in what the remainder the angle makes with  revolutions:

, therefore our angle lies in the fourth quadrant.

Alternatively, we could find evaluate  and .

The former (sine) gives us a negative number whereas the latter (cosine) gives a positive. The only quadrant in which sine is negative and cosine is positive is the fourth quadrant.

Step 1: Define the quadrants and the angles that go in:

QI:

QII:

QIII:

QIV:

Step 2: Find the quadrant where  is:

The angle is located in QII (Quadrant II)

First, using the unit circle, we can see that the denominator has a four in it, which means it is a multiple of .

We want to reduce the angle down until we can visualize which quadrant it is in. You can subtract  away from the angle each time (because that is just one revolution, and we end up at the same spot).

If you subtract away  twice, you are left with , which is in quadrant I. 

.

The second quadrant contains angles between  and , plus those with additional multiples of .  The angle  is, after subtracting , is simply , which puts it in the second quadrant.

This problem relies on understanding reference angles and coterminal angles. A reference angle  for an angle  in standard position is the positive acute angle between the x axis and the terminal side of the angle . A table of reference angles for each quadrant is given below.

Since  is negative, solutions for  will be in Quadrants II and III because these are the quadrants where cosine is negative.

Use inverse cosine and a calculator to find :

In Quadrant II, we have , so .

In Quadrant III, , so . 

Therefore  and .

This problem relies on understanding reference angles and coterminal angles. A reference angle  for an angle  in standard position is the positive acute angle between the x axis and the terminal side of the angle . A table of reference angles for each quadrant is given below.

Since  is negative, solutions for  will be in Quadrants II and IV because these are the quadrants where tangent is negative. Use inverse tangent and a calculator to find :

In Quadrant II, we have ,  so .

In Quadrant IV, , so .

Therefore and .

We can use reference angles, inverse trig, and a calculator to solve this problem. Below is a table of reference angles. 

We have  so . Next, think about where sine is negative, or reference the Function Signs column of the above table. Sine is negative in Quadrants III and IV.

In Quadrant III, .

In Quadrant IV, .

If this problem asked for values of  between  and , our work would be done, but this problem does not restrict the range, so we need to give all possible values of  by generalizing our answers. To do this, we must understand that all angles that are coterminal to  and  will also be solutions. Coterminal angles add or subtract multiples of . To write this generally, we write:

 and .