Circle Arc To Radians: Finding The Angle Range

Hey math enthusiasts! Ever wondered how to convert degree measures of arcs into radians and figure out which range they fall into? Today, we're diving deep into a problem that'll test your understanding of circle geometry and angle conversions. We've got an arc on a circle that measures a neat $85^{\circ}$. Your mission, should you choose to accept it, is to determine the measure of the central angle in radians and pinpoint which of the given ranges it belongs to. Let's break this down, shall we?

First off, let's talk about arcs and central angles. In a circle, an arc is just a portion of the circle's circumference. A central angle is an angle whose vertex is the center of the circle, and its sides are radii intersecting the circle at two points. The measure of a central angle is equal to the measure of its intercepted arc. So, if our arc is $85^{\circ}$, our central angle is also $85^{\circ}$. Easy peasy, right? The real challenge comes when we need to switch gears from degrees to radians.

Now, why radians, you ask? Radians are a way to measure angles that's super useful in calculus and higher-level math because it simplifies many formulas. The key relationship to remember is that a full circle, which is $360^{\circ}$, is equivalent to $2 pi$ radians. This is our golden ticket for conversion! To convert degrees to radians, we use the conversion factor $\frac{\pi \text{ radians}}{180^{\circ}}$. So, to convert $85^{\circ}$ to radians, we multiply $85^{\circ}$ by this factor:

$$85^{\circ} \times \frac{\pi \text{ radians}}{180^{\circ}}$$

Let's simplify this fraction. Both 85 and 180 are divisible by 5.

$85 \div 5 = 17$ $180 \div 5 = 36$

So, the measure of our central angle in radians is $\frac{17 pi}{36}$ radians.

Now, the final step is to figure out which range this value falls into. We have four options:

A. 0 to $\frac{\pi}{2}$ radians B. $\frac{\pi}{2}$ to $\pi$ radians C. $\pi$ to $\frac{3 \pi}{2}$ radians D. $\frac{3 \pi}{2}$ to $2 pi$ radians (Note: The original prompt had a typo for option D, it should be to $2 pi$ radians for a full circle context, but sticking to the prompt's options, let's assume it was intended as rac{3 pi}{2} to $2 pi$)

Let's analyze these ranges.

Range A: 0 to $\frac{\pi}{2}$ radians This range represents angles between $0^{\circ}$ and $90^{\circ}$. Since our angle is $85^{\circ}$, it's definitely less than $90^{\circ}$ (or $\frac{\pi}{2}$ radians). So, this looks like a strong contender!

Range B: $\frac{\pi}{2}$ to $\pi$ radians This range corresponds to angles between $90^{\circ}$ and $180^{\circ}$. Our $85^{\circ}$ angle is less than $90^{\circ}$, so it doesn't fit here.

Range C: $\pi$ to $\frac{3 \pi}{2}$ radians This range is for angles between $180^{\circ}$ and $270^{\circ}$. Clearly, $85^{\circ}$ is way too small for this.

Range D: $\frac{3 \pi}{2}$ to $2 pi$ radians This range covers angles from $270^{\circ}$ to $360^{\circ}$. Again, $85^{\circ}$ doesn't fit here.

Based on our degree measurement, $85^{\circ}$ clearly falls between $0^{\circ}$ and $90^{\circ}$. In radians, this is the range from 0 to $\frac{\pi}{2}$.

Let's double-check using our calculated radian value: $\frac{17 pi}{36}$.

We need to compare $\frac{17 pi}{36}$ with the boundaries of our ranges.

• Is $\frac{17 pi}{36}$ greater than 0? Yes, it is.
• Is $\frac{17 pi}{36}$ less than $\frac{\pi}{2}$? To compare these, we can find a common denominator. $\frac{\pi}{2}$ is the same as $\frac{18 pi}{36}$. Since $17 < 18$, it means $\frac{17 pi}{36}$ is indeed less than $\frac{18 pi}{36}$ (or $\frac{\pi}{2}$).

So, $\frac{17 pi}{36}$ radians is between 0 and $\frac{\pi}{2}$ radians.

This means our answer is Option A.

It's super important to nail these conversions, guys. Understanding that $360^{\circ}$ equals $2 pi$ radians is the fundamental piece of information you need. From there, it's just a matter of setting up the correct ratio for conversion. The factor $\frac{\pi}{180}$ is your best friend when going from degrees to radians, and $\frac{180}{\pi}$ is what you use if you're going the other way around. Always remember that $\pi$ radians is $180^{\circ}$ (a straight line) and $\frac{\pi}{2}$ radians is $90^{\circ}$ (a right angle). These benchmarks are incredibly helpful for quickly estimating where an angle will fall.

Think of the unit circle as your playground. The first quadrant goes from 0 to $\frac{\pi}{2}$ radians. The second quadrant goes from $\frac{\pi}{2}$ to $\pi$. The third quadrant goes from $\pi$ to $\frac{3 \pi}{2}$, and the fourth quadrant goes from $\frac{3 \pi}{2}$ to $2 pi$. Our angle of $85^{\circ}$ is less than $90^{\circ}$, so it's cozy in the first quadrant.

Let's recap the steps to make sure this is crystal clear:

1. Identify the given information: We have an arc measure of $85^{\circ}$. Remember, the central angle equals the arc measure.
2. Recall the conversion factor: $360^{\circ} = 2 pi$ radians, which simplifies to $180^{\circ} = pi$ radians.
3. Convert degrees to radians: Multiply the degree measure by $\frac{\pi}{180}$. So, $85^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{85 pi}{180} = \frac{17 pi}{36}$ radians.
4. Determine the range: Compare the radian measure to the boundaries of the given ranges. We found that $\frac{17 pi}{36}$ is between 0 and $\frac{\pi}{2}$ radians.

So, the measure of the central angle in radians is within the range of 0 to $\frac{\pi}{2}$ radians. It's awesome how these concepts connect, right? Keep practicing these conversions, and soon you'll be a pro at navigating between degrees and radians! If you ever get stuck, just visualize the unit circle and those key radian measures like $\frac{\pi}{2}$, $\pi$, and $\frac{3 \pi}{2}$. They're like your trusty landmarks on the math map. Keep up the great work, everyone!