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Trigonometry : Angles in Different Quadrants

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Example Questions

Example Question #61 : Angles

The angle $\small \frac{-3\pi}{2}$ divides which two quadrants?

Possible Answers:

IV and III

II and IV

I and II

II and III

I and IV

Correct answer:

I and II

Explanation:

$\small \frac{-3\pi}{2}$ is coterminal with the angle $\frac{\pi}{2}$ , or 90^o . This splits quadrants I and II:

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Example Question #522 : Trigonometry

Which angle is not in quadrant III?

Possible Answers:

$\small \frac{-8 \pi}{3 }$

$\small \frac{-11 \pi}{4 }$

$\small \frac{7 \pi}{3}$

$\small \frac{5 \pi}{4 }$

$\small \frac{19 \pi}{6 }$

Correct answer:

$\small \frac{7 \pi}{3}$

Explanation:

First lets identify the angles that make up the third quadrant. Quadrant three is $180^\circ$ to $270^\circ$ or in radians, $\pi$ to $\frac{3\pi}{2}$ thus, any angle that does not fall within this range is not in quadrant three.

Therefore, the correct answer,

$\small \frac{7\pi}{3}$ is not in quadrant three because it is in the first quadrant.

This is clear when we subtract

$\small \frac{7\pi}{3} - 2 \pi = \frac{7 \pi}{3} - \frac{6\pi}{3} = \frac{\pi }{3}$ .

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Example Question #13 : Angles In Different Quadrants

Which two angles are both in the same quadrant?

Possible Answers:

$\small \frac{13\pi}{4}$ and $\small \frac{-\pi}{6}$

$\small \frac{-4\pi}{3}$ and $\small \frac{9\pi}{4}$

$\small \frac{-2\pi}{3}$ and $\small \frac{13\pi}{4}$

$\small \small \frac{11\pi}{6}$ and $\small \frac{\pi}{4}$

$\small \frac{-2\pi}{3}$ and $\frac{-4\pi}{3}$

Correct answer:

$\small \frac{-2\pi}{3}$ and $\small \frac{13\pi}{4}$

Explanation:

First lets identify the different quadrants.

Quadrant I: $0\rightarrow \frac{\pi}{2}$

Quadrant II: $\frac{\pi}{2}\rightarrow \pi$

Quadrant III: $\pi \rightarrow \frac{3\pi}{2}$

Quadrant IV: $\frac{3\pi}{2}\rightarrow 2\pi$

Now looking at our possible answer choices, we will add or subtract $2\pi$ until we get the reduced fraction of the angle. This will tell us which quadrant the angle lies in.

$\small \frac{-2\pi}{3}+2\pi=\frac{-2\pi}{3}+\frac{6\pi}{3}=\frac{4\pi}{3}$ thus in quadrant III.

$\small \frac{13\pi}{4}-2\pi=\frac{13\pi}{4}-\frac{8\pi}{4}=\frac{5\pi}{4}$ thus in quadrant III.

Therefore,

$\small \frac{-2\pi}{3}$ and $\small \frac{13\pi}{4}$ is the correct answer.

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Example Question #72 : Angles

Which angle is in quadrant II?

Possible Answers:

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

$\small \frac{-5\pi}{3}$

$\small \frac{-\pi}{4}$

$\small \frac{-2\pi}{3}$

$\small \frac{-\pi}{2}$

Correct answer:

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

Explanation:

First lets identify the different quadrants.

Quadrant I: $0\rightarrow \frac{\pi}{2}$

Quadrant II: $\frac{\pi}{2}\rightarrow \pi$

Quadrant III: $\pi \rightarrow \frac{3\pi}{2}$

Quadrant IV: $\frac{3\pi}{2}\rightarrow 2\pi$

The correct answer, $\small \frac{-4\pi}{3}$ , is coterminal with $\small \frac{2\pi}{3}$ .

We can figure this out by adding $\small 2\pi$ , or equivalently $\small \frac{6\pi}{3}$ to get $\small \frac{2\pi}{3}$ , 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.

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Example Question #531 : Trigonometry

In which angle would a 50000^o angle terminate in?

Possible Answers:

Quadrant I

Quadrant II

Quadrant III

Between quadrants

Quadrant IV

Correct answer:

Quadrant IV

Explanation:

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 50000^o makes 138 full revolutions. Now the key lies in what the remainder the angle makes with 138 revolutions:

50000^o - 138 * 360^o = 320^o

270^o < 320^o < 360^o , therefore our angle lies in the fourth quadrant.

Alternatively, we could find evaluate $sin \ 50000^o$ and $cos \ 50000^o$ .

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.

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Example Question #71 : Angles

Which quadrant does $135^\circ$ belong?

Possible Answers:

III

Correct answer:

Explanation:

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

QI:

$0^\circ<x\leq 90^\circ$

QII:

$90^\circ < x \leq 180^\circ$

QIII:

$180^\circ < x \leq 270^\circ$

QIV:

$270^\circ < x \leq 360$

Step 2: Find the quadrant where $135^\circ$ is:

The angle is located in QII (Quadrant II)

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Example Question #72 : Angles

The angle $\theta=17\pi/4$ is in which quadrant?

Possible Answers:

Quadrant IV

Quadrant III

Quadrant I

Quadrant II

Correct answer:

Quadrant I

Explanation:

First, using the unit circle, we can see that the denominator has a four in it, which means it is a multiple of $\pi/4$ .

We want to reduce the angle down until we can visualize which quadrant it is in. You can subtract $2\pi$ away from the angle each time (because that is just one revolution, and we end up at the same spot).

If you subtract away $2\pi$ twice, you are left with $\pi/4$ , which is in quadrant I.

$17\pi/4-8\pi/4-8\pi/4=\pi/4$ .

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Example Question #73 : Angles

Which of the following angles lies in the second quadrant?

Possible Answers:

$\frac{11\pi }{4}$

$\frac{\pi }{3}$

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

$\frac{7\pi }{2}$

$\frac{11\pi }{6}$

Correct answer:

$\frac{11\pi }{4}$

Explanation:

The second quadrant contains angles between $\frac{\pi }{2}$ and $\pi$ , plus those with additional multiples of $2\pi$ . The angle $\frac{11\pi }{4}$ is, after subtracting $2\pi$ , is simply $\frac{3\pi }{4}$ , which puts it in the second quadrant.

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Trigonometry : Angles in Different Quadrants

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #61 : Angles

Example Question #522 : Trigonometry

Example Question #13 : Angles In Different Quadrants

Example Question #72 : Angles

Example Question #531 : Trigonometry

Example Question #71 : Angles

Example Question #72 : Angles

Example Question #73 : Angles

All Trigonometry Resources

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