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Trigonometry : Trigonometry

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #581 : Trigonometry

True or False: You have a sector of a circle and are given the radius of the circle and the total area of the circle. You are able to find the area of the sector.

Possible Answers:

True

False

Correct answer:

False

Explanation:

The formula for finding the area of a circle is $A = \pi r^2$ . If we know the radius and the area of the entire circle, we still do not know the angle that forms the sector or the arc length. Therefore we do not have enough information to solve for the sector area. In order to find the area of the sector we need to either know the angle of the sector and the radius, or have some means to solve for this information.

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Example Question #7 : Area Of A Sector

A circle has a sector formed by radii of length 3 and an angle of $\frac{5\pi}{4}$ . What is the area of the sector?

Possible Answers:

$A = \frac{9\pi}{8}$

$A = \frac{9\pi}{4}$

$A = \frac{45\pi}{8}$

$A = \frac{8\pi}{5}$

Correct answer:

$A = \frac{45\pi}{8}$

Explanation:

The formula to solve for the area of a sector is $A = \frac{\theta}{2} r^2$ . We have all the information needed to plug the values right into the formula and solve for the area.

$A = \frac{\theta}{2} r^2$

$A = (\frac{5\pi}{4})(\frac{1}{2})(3^2)$

$A = (\frac{5\pi}{8})(9)$

$A = \frac{45\pi}{8}$

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Example Question #8 : Area Of A Sector

You are given a circle with a sector that is formed by an arc length of $9\pi$ . The circle has a diameter of 20. What is the area of the sector?

Possible Answers:

$A=\frac{9\pi}{20}$

$A=\frac{9\pi}{4}$

$A = \frac{9\pi}{2}$

$A=9\pi$

Correct answer:

$A = \frac{9\pi}{2}$

Explanation:

First, if we have a diameter of 20, we know that the radius must be half of the diameter. We have a radius length of 10. Now, we know the arc length so we must use the arc length formula to solve for the angle measure. To find the length of an arc of a sector we use the formula $l_A = \theta r$ .

$l_A = \theta r$

$9\pi = \theta 10$

$\frac{9\pi}{10} = \theta$

By solving for the angle, we have enough information to solve for the sector area.

$A = \frac{\theta}{2} r^2$

$A = (\frac{9\pi}{10})(\frac{1}{2})(10^2)$

$A = (\frac{9\pi}{20}(100)$

$A = \frac{900\pi}{200}$

$A = \frac{9\pi}{2}$

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Example Question #1 : Arc Length

Which is true of the relationship between the arc measure and the central angle as shown below?

Screen shot 2020 08 27 at 4.09.48 pm

Possible Answers:

The central angle will always be a right angle

They are equal

The central angle is half of the arc length

The arc length is half of the central angle

Correct answer:

They are equal

Explanation:

Every arc has a measure that is equal to the measure of the central angle that creates the arc. This is because the measure of the angle determines the distance around the circumference that the arc makes.

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Example Question #2 : Arc Length

Which of the following is the correct formula for finding arc length?

Possible Answers:

$l_A = \theta r^2$

$l_A = 2\theta r$

$l_A = \theta r$

$l_A = 2\theta r^2$

Correct answer:

$l_A = \theta r$

Explanation:

The circumference of an entire circle is $2\pi r$ . When considering the length of an arc, the angle is less than $2\pi$ denoted by angle $\theta$ . So the formula for finding the length of an arc is replacing the angle of an entire circle, $2\pi$ , with the angle that forms the arc, $\theta$ . This gives us the formula $l_A = \theta r$ .

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Example Question #3 : Arc Length

Which of the following is the correct arc length formed by the angle $\frac{\pi}{3}$ of a circle whose radius is a length of 5?

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

We must use the formula for finding arc length $l_A = \theta r$ . We have been given all the information needed to just plug into the formula.

$l_A = \theta r$

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

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

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Example Question #4 : Arc Length

Which of the following is the correct arc length formed by an angle with measure 30 degrees of a circle whose radius is a length of 3?

Possible Answers:

$l_A = 2\pi$

$l_A = \frac{\pi}{2}$

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

$l_A = \frac{\pi}{6}$

Correct answer:

$l_A = \frac{\pi}{2}$

Explanation:

First, we are given our angle measure in degrees and we must convert to radians to be able to use our arc length formula.

$30 \frac{\pi}{180} = \frac{\pi}{6}$

Now we are able to plug the radius length and the angle measure into our formula and solve for the arc length.

$l_A = \frac{\pi}{6} 3$

$l_A = \frac{3\pi}{6}$

$l_A = \frac{\pi}{2}$

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Example Question #5 : Arc Length

What is the measure of the angle that forms an arc with length 2.33 of a circle who has radius 4? Round to the second decimal place.

Possible Answers:

$\theta = 0.58$

$\theta = 2.33$

$\theta = 1.43$

$\theta = 4.00$

Correct answer:

$\theta = 0.58$

Explanation:

We must use the formula for finding arc length to solve for the measure of the angle, $\theta$ . The formula is $l_A = \theta r$ .

$l_A = \theta r$

$2.33 = \theta 4$

$0.58 = \theta$

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Example Question #2 : Arc Length

An arc has a measure of $\frac{\pi}{4}$ and a diameter of 7, what is the measure of the central angle?

Possible Answers:

$\frac{\pi}{4}$

$\frac{\pi}{6}$

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

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

Correct answer:

$\frac{\pi}{4}$

Explanation:

Notice the question gave you the measure of the arc, NOT the arc length. The measure of an arc is equal to the measure of the central angle that forms the arc. We do not even need to use our formula for this one, just the fact that the central angle is equal to the measure of the arc that it forms.

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Example Question #3 : Arc Length

True or False: You are given the central angle measure but no other information, you are able to solve for the arc length.

Possible Answers:

True

False

Correct answer:

False

Explanation:

We do not have enough information to solve for the arc length. We know that the measure of the arc is equal to the central angle and that the measure of the arc is $\theta = \frac{r}{l_A}$ but we still have two unknowns with no method to solve for them.

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Trigonometry : Trigonometry

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #581 : Trigonometry

Example Question #7 : Area Of A Sector

Example Question #8 : Area Of A Sector

Example Question #1 : Arc Length

Example Question #2 : Arc Length

Example Question #3 : Arc Length

Example Question #4 : Arc Length

Example Question #5 : Arc Length

Example Question #2 : Arc Length

Example Question #3 : Arc Length

All Trigonometry Resources

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