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Trigonometry : Unit Circle and Radians

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Example Questions

Example Question #11 : Unit Circle

Suppose there is exists an angle, $\theta = \frac{n\pi }{x}{}$ such that $cos \ \theta = \frac{1}{2}$ .

For what values of and make this trigonometric ratio possible?

Possible Answers:

$x = 3 \ or \ 6$

n = 1, 5, 11

x = 3

$n = 1 \ or \ 5$

x = 6

$n = 1 \ or \ 11$

x = 6

n = 1

x = 3

n = 1

Correct answer:

x = 3

$n = 1 \ or \ 5$

Explanation:

The only values such that

$cos \ \theta = \frac{1}{2}$

are at the values:

$\theta = \frac{\pi}{3} , \frac{5 \pi }{3}$

This means that the only choice for is . n = 1 or $n = \ 5$ achieve the necessary angles to satisfy this trigonometric ratio.

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Example Question #11 : Unit Circle

If $tan\left ( \theta \right )=\frac{3}{4}$ , and $\pi \leq \theta \leq \frac{3\pi}{2}$ , what is $sin(\theta)$ ?

Possible Answers:

$-\frac{4}{5}$

$-\frac{3}{4}$

$\frac{4}{5}$

$\frac{3}{5}$

$-\frac{3}{5}$

Correct answer:

$-\frac{3}{5}$

Explanation:

The tangent of an angle yields the ratio of the opposite side to the adjacent side.

If this ratio is $\frac{3}{4}$ , we can see that this is a Pythagorean triple (3-4-5); the absolute value of the sine of this angle would be $\frac{3}{5}$ .

However, the question indicates that this angle lies in the 3rd quadrant. The sine of any angle in the 3rd or 4th quadrant is negative, since it is equivalent to the y-coordinate of the corresponding point on the unit circle.

Therefore,

$sin(\theta)=-\frac{3}{5}$ .

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Example Question #11 : Unit Circle And Radians

If $cos(\phi)=-\frac{1}{2}$ , which of the following angles is NOT a possible value for $\phi$ ?

Possible Answers:

$480^{\circ}$

$300^{\circ}$

$120^{\circ}$

$-120^{\circ}$

$240^{\circ}$

Correct answer:

$300^{\circ}$

Explanation:

On the unit circle, the cosine of an angle yields the x-coordinate.

There are two angles at which the x-coordinate on the unit circle is $-\frac{1}{2}$ : $120^{\circ}$ and $240^{\circ}$ .

$-120^{\circ}$ is coterminal with $240^{\circ}$ , and $480^{\circ}$ is coterminal with $120^{\circ}$ .

$300^{\circ}$ is in the 4th quadrant, and has a positive x-coordinate.

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Example Question #11 : Unit Circle

How many degrees are in a unit circle?

Possible Answers:

360

240

720

180

Correct answer:

360

Explanation:

Step 1: Define a Unit Circle:

A unit circle is used in Trigonometry to draw and describe distinct angles. The unit circle works along with the coordinate grid.

Step 2: There are quadrants in the coordinate grid, each quadrant can fit degrees.

Step 3: Multiply how many degrees in quadrant by .. to get the full unit circle:

$90 \cdot 4=360$

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Example Question #11 : Unit Circle And Radians

Change $\frac{4\pi }{3}$ angle to degrees.

Possible Answers:

$300^{\circ}$

$240^{\circ}$

$180^{\circ}$

$150^{\circ}$

Correct answer:

$240^{\circ}$

Explanation:

In order to change an angle into degrees, you must multiply the radian by $\frac{180^{\circ}}{\pi}$ .

Therefore, to solve:

$\frac{4\pi }{3} \cdot \frac{180^{\circ} }{\pi} = 240^{\circ}$

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Example Question #1 : Radians And Conversions

Give in radians:

$\frac{x+\pi}{x-\pi}=30^{\circ}$

Possible Answers:

$x=\frac{\pi(\pi+8)}{\pi-8}$

$x=\frac{\pi(\pi+6)}{\pi-6}$

$x=\frac{\pi(\pi-6)}{\pi+6}$

$x=\frac{\pi+6}{\pi-6}$

$x=\pi$

Correct answer:

$x=\frac{\pi(\pi+6)}{\pi-6}$

Explanation:

First we need to convert degrees to radians by multiplying by $\frac{\pi}{180^{\circ}}$ :

$30^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{6}$

Now we can write:

$\frac{x+\pi}{x-\pi}=\frac{\pi}{6}\Rightarrow 6(x+\pi)=\pi(x-\pi)$

$\Rightarrow 6x+6\pi=x\pi-\pi^2\Rightarrow 6x-x\pi=-6\pi-\pi^2$

$\Rightarrow x(6-\pi)=-\pi(\pi+6)$

$\Rightarrow x=\frac{-\pi(\pi+6)}{6-\pi}=\frac{\pi(\pi+6)}{\pi-6}$

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Example Question #171 : New Sat Math Calculator

Give in radians:

$x-\pi=120^{\circ}$

Possible Answers:

$x=\frac{5\pi}{6}$

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

$x=\frac{\pi}{3}$

$x=\frac{2\pi}{3}$

$x=\frac{5\pi}{3}$

Correct answer:

$x=\frac{5\pi}{3}$

Explanation:

First we need to convert degrees to radians by multiplying by $\frac{\pi}{180^{\circ}}$ :

$120^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{3}$

Now we can write:

$x-\pi=120^{\circ}\Rightarrow x-\pi=\frac{2\pi}{3}\Rightarrow x=\frac{2\pi}{3}+\pi=\frac{5\pi}{3}$

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Example Question #4 : Radians

Give in degrees:

$\frac{3x+6\pi}{2}=4\pi$

Possible Answers:

$x=150^{\circ}$

$x=210^{\circ}$

$x=30^{\circ}$

$x=120^{\circ}$

$x=180^{\circ}$

Correct answer:

$x=120^{\circ}$

Explanation:

First we can find in radians:

$\frac{3x+6\pi}{2}=4\pi\Rightarrow 3x+6\pi=2\times 4\pi$

$\Rightarrow 3x+6\pi=8\pi\Rightarrow 3x=8\pi-6\pi$

$\Rightarrow 3x=2\pi\Rightarrow x=\frac{2\pi}{3}$

To change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ . So we can write:

$x=\frac{2\pi}{3}\Rightarrow x=\frac{2\pi}{3}\times \frac{180^{\circ}}{\pi}=120^{\circ}$

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Example Question #5 : Radians

Give in degrees:

$\frac{4x-\pi}{2}=3\pi$

Possible Answers:

$x=315^{\circ}$

$x=305^{\circ}$

$x=300^{\circ}$

$x=345^{\circ}$

$x=415^{\circ}$

Correct answer:

$x=315^{\circ}$

Explanation:

First we can find in radians:

$\frac{4x-\pi}{2}=3\pi\Rightarrow 4x-\pi=2\times 3\pi$

$\Rightarrow 4x-\pi=6\pi\Rightarrow 4x=6\pi+\pi$

$\Rightarrow 4x=7\pi\Rightarrow x=\frac{7\pi}{4}$

To change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ . So we can write:

$x=\frac{7\pi}{4}\Rightarrow x=\frac{7\pi}{4}\times \frac{180^{\circ}}{\pi}=315^{\circ}$

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Example Question #3 : Radians

Give in radians:

$\frac{72^{\circ}-\pi}{2x-\pi}=3$

Possible Answers:

$x=\frac{\pi}{5}$

$x=\frac{2\pi}{3}$

$x=\frac{3\pi}{5}$

$x=\frac{2\pi}{5}$

$x=\frac{4\pi}{5}$

Correct answer:

$x=\frac{2\pi}{5}$

Explanation:

First we need to convert $72^{\circ}$ to radians by multiplying by $\frac{\pi}{180^{\circ}}$ :

$72^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{5}$

Now we can solve the following equation for :

$\frac{72^{\circ}-\pi}{2x-\pi}=3\Rightarrow \frac{\frac{2\pi}{5}-\pi}{2x-\pi}=3$

$\Rightarrow {\frac{2\pi}{5}-\pi}=3(2x-\pi)\Rightarrow \frac{2\pi-5\pi}{5}=6x-3\pi$

$\Rightarrow -3\pi=30x-15\pi\Rightarrow 30x=12\pi$

$\Rightarrow x=\frac{12\pi}{30}\Rightarrow x=\frac{2\pi}{5}$

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Trigonometry : Unit Circle and Radians

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #11 : Unit Circle

Example Question #11 : Unit Circle

Example Question #11 : Unit Circle And Radians

Example Question #11 : Unit Circle

Example Question #11 : Unit Circle And Radians

Example Question #1 : Radians And Conversions

Example Question #171 : New Sat Math Calculator

Example Question #4 : Radians

Example Question #5 : Radians

Example Question #3 : Radians

All Trigonometry Resources

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