Trigonometric Functions – Angles and Their Measure | A denotes the area of the sector of a circle

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Trigonometry: Finding the Central Angle from the Area of a Sector

This lesson begins the trigonometry problem solving portion of the crash course series. The example focuses on applying the area of a sector formula to determine a missing central angle measured in radians. :contentReference[oaicite:0]{index=0}

The Problem

Given:

Radius: 5 miles
Sector Area: 3 square miles
Central Angle: Unknown

Find the missing angle and round the answer to three decimal places. :contentReference[oaicite:1]{index=1}

Step 1: Recall the Sector Area Formula

The area of a sector is given by:

$A=\frac{1}{2}r^2\theta$

where:

$A$ = area of the sector
$r$ = radius
$\theta$ = angle in radians

This formula must be memorized because it appears repeatedly throughout trigonometry, calculus, and physics. :contentReference[oaicite:2]{index=2}

Students entering calculus should become comfortable recalling important trigonometric formulas directly from memory.

Step 2: Solve the Formula for Theta

Starting with:

$A=\frac{1}{2}r^2\theta$

Multiply both sides by 2:

$2A=r^2\theta$

Now divide by $r^2$ :

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

Step 3: Substitute the Known Values

Insert the given measurements:

$\theta=\frac{2(3)}{(5)^2}$

Simplify:

$\theta=\frac{6}{25}$

Converting to decimal form:

$\theta=0.24$

Step 4: Final Answer

Rounded to three decimal places:

$\theta=0.240\text{ radians}$

Therefore:

$\boxed{\theta=0.240\text{ radians}}$

Why the Units Matter

The lesson emphasizes the importance of carrying units throughout the calculation.

Notice that:

The area contains square miles.
The radius squared also contains square miles.
The square mile units cancel.
The result is left in radians.

Failing to track units can lead to lost points on examinations and mistakes in applied mathematics and physics problems. :contentReference[oaicite:3]{index=3}

The Bigger Picture

This example demonstrates a recurring theme throughout trigonometry: many problems begin with a memorized formula, require algebraic manipulation, and conclude with careful substitution and unit analysis.

The specific numerical answer is important, but understanding the formula and knowing when to apply it is what ultimately prepares students for calculus, differential equations, engineering, and physics. :contentReference[oaicite:4]{index=4}

Final Thoughts

Students often focus exclusively on answers, but successful problem solving depends on understanding the formulas, carrying units correctly, and showing a complete mathematical process. As mathematics becomes more advanced, the setup and justification become increasingly important and often earn the majority of the credit on exams. :contentReference[oaicite:5]{index=5}

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Original Transcript

Kids, let’s continue on our problem solving journey. Here we are entering into the trigonometry world now. And uh trigonometry [clears throat] starts here for chapter seven in this book, but it’s actually chapter one in the in the regular book. So I’m going to title this one college trigonometry. Trigonometry. This will be heading one. And then section seven this will be um chapter 1 trigonometric functions and then we are going to go to section angles and their measurements. So this one will be heading two and this one will be heading one three sorry and we will take a question from this book here which is going to be so I do question and this is book number 19 so book 19 and then it is [clears throat] I guess I’ll I guess I’ll leave it as chapter seven for now or uh well I don’t really want to label it chapters per se because I’m just taking questions from random random books. So I’m I’m going to leave it like that and then the reference will be there. So then this is 7.1 we’ll find a question here now to use and uh we’ll start off basic. I’m going to go through essentially each one of these each one of these um problem sets. So they want to draw each angle and uh convert each angle to a decimal in degrees. Convert each angle divert each angle in degrees to radians. Express your answer as a multiple of pi. So I’m going to go through and do like each one of these probably from every book for the most part. from each thing. So let’s start with uh we’ll start with a I don’t I don’t know these these these are too easy. Let’s let’s move on to something a little bit more. A denotes the area of the sector. I’m going to I’m going to choose a problem from every single section. This is a crash course. I’m not going to do everything. So I am going to take this question here and let me let me actually get the screenshot of the question and I can type it in.

So I am going to use

let’s see so a denotes the area of the sector of the circle of radius r that should be in the equation editor at the central angle theta find the missing quantity round the answers to three decimal places so I’m going to do question this is 7.1 one dot uh guess I’ll do 83 question 83 and that question is r is equal to 5 miles and a is equal to 3 square miles and theta is equal to question mark so let’s make sure that’s right 5 miles 3 square miles theta is question mark. Good. And now we’ll do the answer section.

I just divide it like this. You can if you’re typing you can divide it however you want. This was 7.1.83. I’m going to go down to answers now.

7.1.83.

So 7.1.

See if we can jump to it a little faster here. There’s a lot of 7.1s. If I hit space, maybe. There we go. 7.1. Oops. Oopsy. I went right past it. There we go. Go for it. 7.1 number 83 is 0.24 radian. Okay. So 0.24 radian. Okay. Let’s uh answer these questions. I’m doing in this book. I’m answering it the way you would like on a test. So,

so A is the area A is the area of the sector of a circle of radius R formed by the central angle theta. Find the missing quantity. Round your answers to three decimal places. Okay, so we got three square miles is the area of a sector and then we have the radius is 5 miles. So we should probably get the formula for a sector first. So on a test you would have to take this out of memory. So I’m going to go to the section and type sector and get the formula. So area of a sector is a is equal to 12 pi 12 r^2 theta. So area of sector

is one is a is equal to 12 r 2 theta. Are there conditions on that?

The area of a sector of a circle radius r formed by the central angle r is equal to 12 r a. Okay. Okay. So that’s that’s our that’s our citation. Now this is then going to be solved to be r^ squared is equal to 2 a over theta. All right. So 2 a over theta and then we can take the square root of that. r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r is always greater than zero. Okay, great. R is greater than zero.

So I should I’ll do this like that and then do therefore r is equal to

the square t of 2 a over theta. All right. So therefore, r is equal to the square t of

2 * 3 m^ 2 over 5 miles 3 square miles over 5 miles. And for the units, that’s going to be the same as saying squared here. And uh so I can then make this

that’s supposed to be miles.

[snorts] If you guys don’t pay attention to the units, you will lo you will miss points on the exam big time. Okay. So then that becomes the square of 6 over 5.

And

wait, what are we looking? We’re looking to solve for for theta. Hold on. [laughter] Okay, hold on. I did that wrong. I was like, something’s not right with these units here. This is um Let’s go back here. This is theta. Theta is 2 a over r 2. And so r we should acknowledge is greater than zero. And uh that’s fine. That that’s fine. So I was I was solving for r in my Okay. Now therefore

theta is equal to 2 a over r squared. And this is where you want to pay attention to the units.

And so then this is the same as saying

miles squared. And so now we’re going to get six fifths six fifths because the miles will cross out. So if it if it helps you to uh write more, you can if you want to do two C do 3 / 5 here and then take the miles squared like so.

And then you can see that those cross out.

Okay. So then we’re left with six six fths radian. And 6 fths is 6 fths.24. I don’t know. I’m not an arithmetic. I’m a mathematician. So I will uh say that’s 6 fths is uh definitely not 2.0.24. Right. [laughter] That was so that the question is though we use we have miles and area 53. Did I do this right? I forgot to square the uh 25. That’s where I messed up. That’s where I messed up. So this should be 25. Little tiny typo. So then 6 over 25. That sounds like it’s probably 0.24. Okay. 0.24.

Now it says round answers to three decimal places. That’s only two. [laughter] Uh, the question says to round to three decimals. I may be inclined to say 0.240 radian

in in a physics course for significant figures. But th this is how you would do this on a test. Okay? And then I’ll just put like a at the end of this for you guys notes. Just a note for y’all. Just a noty note. Okay. Okay. So, we got that answer correct as far as we can tell. This part here is 50% of the credit on the test. This part here’s the other 50%. The answer is worth nothing. So you say at the very end you say therefore theta is equal to 0.24 radians radian. All right. So that’s uh that’s the first lesson for the trigonometry thing here. Thanks for watching. Thanks for listening. If you guys can join, leave a donation, buy a book, I greatly appreciate it. I do this for a living, so without your support, I cannot do it. Have a nice day.