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Problem 40 Hard Difficulty

In the figure, the length of the chord $ AB $ is $ 4 $ cm and the length of the arc is $ AB $ is $ 5 $ cm. Find the central angle $ \theta $, in radians, correct to four decimal places. Then give the answer to the nearest degree.

Answer

$\theta \approx 2.2622$ radians, or $130^{\circ}$

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Video Transcript

question 40. You start with the drawing. So years. So my are playing. It's five. And my cord is for with this my A It is also my a my radiance. All right, So ate my radius and my cord. A B equals four. So I have, um, data there or five daytime stada A equals five divided by data. I'm gonna use my law of co signs. Which does he squared? Just mike course. Where'd my cortland equals? A squared plus B squared, but a square to be squared with the same evidence. Um, have a squared plus a squared minus to a B, which is to a squared co sign data. So now I'm gonna plug in my four for my C force. Weird equals, and I'm gonna factor out. I'm gonna simplify. I am to a squared minus 2/8. Weird. Co sign they either 16 equals two A's. Clear. One minus co sign. See that eight equals just divided by two. Now I have a square one minus co sign. They, uh I'm going to replace a With what? My A is alive, vibrant leader A equals five over. Data squared one minus. Co sign data in equals 25 over. Fate of sprayer minus. I'm going to get data by itself. Something this weird date of both sides. So I get it. See that squared equals 25 times. What times? What? 25 times? One minus. No, I I got a contract. Eight data from boats. No. Gonna bring over the other. So I multiplied by 25. So I get Hey, they either. Minus 20 bye. Minus 20. No. Plus 25 co sign. Hey, there. That is my f Evita. Just gonna put a box around it that I'm gonna change my f fry data, so I'm gonna have Where? No. The night of the crime. I'm gonna have 16. 16 beta less the finest minus 25 Sign data. That's my money. Okay? No, I have my crime. Now. I can do my Newton's approximation, so I'm going to start with Exit one, which is really Laida of one equals mr with 2.2 from looking at the graph and that I plug in. Remember my s of experts. One for me, um equals minus. So my exit too, for my status of two really is two point 266223 Except three, which is really my state status and three equals two point 26 22 to 0. And that's also my It is a forward. So I stopped there. Um and that is in radiance. I want it in degrees. So I take 2.26222 times. Well, you were. I get they even equal to 1 29 White 62 Agrees. War 1 30 degrees if you're around it.