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Trigonometry : Angle Applications

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : 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:

$\theta = 0.58$

$\theta = 1.43$

$\theta = 4.00$

$\theta = 2.33$

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:

$\frac{\pi}{6}$

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

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

$\frac{\pi}{4}$

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:

False

True

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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Example Question #4 : Arc Length

You know that the central angle of a sector is $\frac{3\pi}{2}$ and the sector area is $\frac{7\pi}{4}$ , what is the arc length? Round to two decimals.

Possible Answers:

l_A = 14.67

l_A = 12.76

l_A = 9.39

l_A = 2.51

Correct answer:

l_A = 12.76

Explanation:

The formula to find the area of a sector is $A = \frac{\theta}{2} r^2$ . The piece of information we are missing to solve for arc length is the radius, so we will use the formula for finding the area of a sector to solve for radius, allowing us to solve for the arc length.

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

$\frac{7\pi}{4} = (\frac{3\pi}){2}(\frac{1}{2})r^2$

$\frac{7\pi}{4} = (\frac{3\pi}{4} r^2$

$\frac{7\pi}{3} = r^2$

$\sqrt{\frac{7\pi}{3}} = r$

2.707 = r

Now we can plug this into our formula to solve for arc length

$l_A = \theta r$

$l_A = \frac{3\pi}{2} (2.707)$

l_A = 12.76

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Trigonometry : Angle Applications

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Arc Length

Example Question #2 : Arc Length

Example Question #3 : Arc Length

Example Question #4 : Arc Length

All Trigonometry Resources

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