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Trigonometry : Unit Circle

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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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Trigonometry : Unit Circle

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

All Trigonometry Resources

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