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Find, correct to the nearest degree, the three angles of the triangle with the given vertices.

$ P (2, 0) $ , $ Q (0, 3) $ , $ R (3, 4) $

$$48^{\circ}, 75^{\circ}, 57^{\circ}$$

03:44

Wen Z.

10:51

Bobby B.

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

Johns Hopkins University

Campbell University

Harvey Mudd College

University of Nottingham

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we have a question in this. Uh We have been given three points and we need to find the value of each angles of the uh tangle made by this point. Split pea. Two comma zero Q. Zero comma three. Our three comma four sort of flea Let us say this is B. This is Q. This is our so length off peak you because we'll be using a sine or cosine law for that length is required. So PQ equal to from the uh what distance formula to minus zero whole square plus zero minus three holds court under road. So four plus nine. Under route 13 Q. R. Will be equal to three minus zero old square. That extra minus 614 minus three on square on the road. So nine plus one. Under 10. An RP will be equal to three minus two. Old square plus four minus zero. Old square for 16 plus 1 17. And the root. So roughly we can write PQ equal to under 13 Q. Are equal to under 10 and are equal to under 17. Okay so right if they take angle B. Of course using cosine law, angle P will be equal to that is cause angle pr Okay Cosby will be cool too. The Qs quiet less P. R squared minus Q. R squared divided by to pick you into P. R. Or R. P. One. And the same thing. Let us plug in the values BQ esquire GQ is under 13 whole square plus pr squarely than the route 17. Whole square minus cure square is under 10. All squared by two into take you. And the route 13. And pr and the route 17 So 13 plus 17 minus 10. Give out by two. On the route 13 to 17. So 30 miles to 30 minutes 10 20 by to win. 213 minus 17. This is done. So we'll be having can buy under the road. 13 to 17. So then divided by Yeah. And the root 13 to 17 0.67 260.67 26. So angle P will be equal to cause university 260.6726. So of course in verse 47.726 47 points 72 63 degrees. Okay this is angle P angle Q. Again, we'll be using the cosign law for angle ah Q. So cause of angle Q will be equal to PQ squared plus Q. R squared PQ esquire plus Cure square minus pr square by two and two PQ into cuba. So this is BQ square. The humans under 13. So under 13 whole square plus Q. R. Square. So under 10 whole square minus pr square. And about 17 whole square divide by divide by two and two under 13 in 210. So this is 13 plus 10 minus 17 by two and 2. 100 30. Okay so this is 23 minus 17 6 by two on the road. One of the 30. So three by and over 30. Okay so three divide by under route 30 which is zero point uh 54. Sorry three D. Were banned at 1 30. This is 0.2631. 0.2631. So you will be equal to cause inverse 0.2631. Yeah. Okay, so 74.7448 74.7448 degrees. Okay. No, since some of all the angles should be 180 degree. So angle are will be angle will be 180 degree minus angle P. And angle Q. Angle page 47.7263 47.7263. And plus this angle angle Q 74.7448 degrees. Okay. Plus 47.47 point 72637263 minus 180 57.5 to 8. So 57.52 80 degrees. Okay, so nearest degree. We have angle P. Approximately angle P. Approximately 48 degrees. 40 degrees, angle Q. Approximately will be 75 degrees. And angle are approximately will be 57 degrees. So these are the answers. Thank you

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