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Trigonometry : Trigonometry

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6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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 #11 : Angles In Different Quadrants

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

Possible Answers:

Quadrant IV

Quadrant II

Quadrant I

Between quadrants

Quadrant III

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 #12 : Angles In Different Quadrants

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 #13 : Angles In Different Quadrants

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

Possible Answers:

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

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 #11 : Angles In Different Quadrants

Which of the following angles lies in the second quadrant?

Possible Answers:

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

$\frac{\pi }{3}$

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

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

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

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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Example Question #1 : Sin, Cos, Tan

Find the value of the trigonometric function in fraction form for triangle ABC .

Triangle

What is the cosine of $\angle B$ ?

Possible Answers:

$\frac{7}{25}$

7/24

24/25

Correct answer:

$\frac{7}{25}$

Explanation:

The cosine of an angle is the value of the adjacent side over the hypotenuse.

Therefore:

$cos \angle B = \frac{adjacent}{hypotenuse} = \frac{7}{25}$

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Example Question #1 : Sin, Cos, Tan

What is the value of sin(30)+sin(60) ?

Possible Answers:

$\frac{\sqrt3+1}{2}$

$\frac{\sqrt3-1}{2}$

Correct answer:

$\frac{\sqrt3+1}{2}$

Explanation:

Solve each term separately.

$cos(30)= \frac{\sqrt3}{2}$

$cos(60)=\frac{1}{2}$

Add both terms.

$\frac{\sqrt3}{2}+\frac{1}{2}= \frac{\sqrt3+1}{2}$

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Example Question #3 : Sin, Cos, Tan

Determine the value of 2tan(120) .

Possible Answers:

$-2\sqrt3$

$-\frac{\sqrt3}{2}$

$2\sqrt3$

$\sqrt3$

$-\sqrt3$

Correct answer:

$-2\sqrt3$

Explanation:

Rewrite 2tan(120) in terms of sines and cosines.

$2tan(120)=2(\frac{sin(120)}{cos(120)})=2(\frac{\frac{\sqrt3}{2}}{-\frac{1}{2}})$

Simplify the complex fraction.

$2(\frac{\frac{\sqrt3}{2}}{-\frac{1}{2}})= 2\times\frac{\sqrt3}{2}\times-2 = -2\sqrt3$

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Example Question #1 : Sin, Cos, Tan

Find the value of $\frac{1}{2}sin(45)+ tan(60)$ .

Possible Answers:

$\frac{\sqrt2-4\sqrt3}{2}$

$\sqrt2+\sqrt3$

$\frac{3\sqrt2+\sqrt3}{12}$

$\frac{\sqrt2+4\sqrt3}{2}$

$\frac{\sqrt2+4\sqrt3}{4}$

Correct answer:

$\frac{\sqrt2+4\sqrt3}{4}$

Explanation:

To find the value of $\frac{1}{2}sin(45)+ tan(60)$ , solve each term separately.

$\frac{1}{2}sin(45)=\frac{1}{2} \cdot \frac{\sqrt2}{2} = \frac{\sqrt2}{4}$

$tan(60) = \sqrt3$

Sum the two terms.

$\frac{\sqrt2}{4}+\sqrt3 = \frac{\sqrt2}{4}+\frac{4\sqrt3}{4} = \frac{\sqrt2+4\sqrt3}{4}$

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Trigonometry : Trigonometry

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #13 : Angles In Different Quadrants

Example Question #72 : Angles

Example Question #11 : Angles In Different Quadrants

Example Question #12 : Angles In Different Quadrants

Example Question #13 : Angles In Different Quadrants

Example Question #11 : Angles In Different Quadrants

Example Question #1 : Sin, Cos, Tan

Example Question #1 : Sin, Cos, Tan

Example Question #3 : Sin, Cos, Tan

Example Question #1 : Sin, Cos, Tan

All Trigonometry Resources

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