Solve each triangle. Approximate values to the nearest tenth.

Question

Accepted Answer

Welcome back. Everyone. In this problem, we want to find the missing measurements of our triangle. For our answer choices. A says angle P is 128.7 degrees. Angle Q is 18.2 degrees and R is angle. Sorry. Yeah, angle R is 33.1 degrees. B says it's 33.1 degrees, 18.2 degrees and 128.7 degrees respectively. C says it's 18.2 degrees, 33.1 degrees and 128.7 degrees respectively. And D says it's 33.1 degrees, 128.7 degrees and 18.2 degrees respectively. Now here in this problem, we're trying to essentially figure out all of the angles in our problem given that we know all of the sides to make it easier to refer to while we work. Let's say that the side opposite to angle P is called P, the side opposite to angle R is called R and the side opposite to anger Q is called Q typically how they label triangles while solving problems. Now, the question is, how can we figure out any of these angles, let's say that we wanted to start with angle P. OK. Then how could we figure angle P out? We know three sides? And we want to find the angle opposite to the side P if we recall the law of core signs. OK. Then in the law of core sciences, it tells us that for any one of the sides, let's say in this case, P squared, then P squared would be equal to Q squared plus R squared minus two QR multiplied by the cosine of P. So notice that in this formula based on the law of cosines, we have been able to include angle P using the three sides of the triangle of which we know are three sides. So we could rearrange our formula then to solve for the cosine of P and thus angle P. So let's go ahead and do that. Let's substitute our values. So we know that P is seven. So that's seven squared, Q is four. So that's four squared and R is 10. So that's 10 squared minus two, multiplied by four, multiplied by 10, multiplied by the cosine of angle P which as we said, we don't know what it is yet. Now, if we continue our work, seven squared is 49 4 squared is 16 and 10 squared is 100 is two multiplied by four, is eight and eight, multiplied by 10 is 80. So that's minus 80 KP here. 16 plus 100 that's 116 minus 80 CSP. And if we add a CSP to both sides and take away 49 then we find that 80 CSP. This is going to be equal to 116 minus 49. So it equal P equals 67. If 80 cos P equals 67 that means cos P is going to be equal to 67 divided by 80. And now to solve for angle P, we can do the inverse of the cosine. In other words, P is going to be equal to the inverse cosine of that expression 67 divided by 80. So now we can find the inverse cosine of 67 divided by 80 in our calculator. And when we do that, OK, we should get an P to be equal to 33.1229 degrees or to the nearest 10th of a degree P is approximately equal to 33.1 degrees. So we found one of the angles in our triangle angle P. So let's put that in our diagram. So so far we know that P is 33.1 degrees. Now, if we think about the information we have now we realize we know three sides and an angle and we want to use that to figure out another angle in our triangle. Notice that angle P is opposite to side P OK. And let's say that we wanted to find angle Q, we know the value of the side opposite to angle Q, we know that the side Q is four units. Now let's ask ourselves if we want to figure out anger Q, OK? If we want to figure out anger Q, and we know a pair of angles and side, how can we figure out the value of Q? What can help us to do so? Well, again, we can use one of the laws and in this case, we can use the sign law. Ok. Recall that the law of science, the law of science basically says that if we have a known pair of anger and a side, then the ratio. Ok? The ratio, well, you don't have to have it but it's it's one of the use cases for it. But the law tells us that the ratio of the sine of an angle. In this case, the sine of P to the side of P is going to be the same ratio as the sine of Q to its opposite side Q, which is the same ratio as the sine of R to its opposite side R. The ratio of the sine of an angle to its opposite side is the same through the triangle. That's what the law of science tells us. Well, we know angle P. OK? Let me take that here. So we know Anger P, we know the length of side P. We don't know what Anger Q is. But we know the length of side Q so we can use the law of science to help us to figure out what that unknown Anger Q is going to be. Because by the law of science, then this means that the sign of Q to its opposite side four is going to be equal to the sine of angle P which we just found to be 33.1 degrees to its opposite side seven. And know that we have that if we multiply both sides of our equation by four, then the sine of Q is going to be equal to four multiplied by the sine of 33.1 degrees divided by seven. And now we can put this expression into our calculator to solve for the sin of Q. Now, when we do that, OK, we should get the sine of Q to be equal to 0.31206. OK? Which then tells us that anger Q H the queue is going to be equal to the inverse sine of that value. So the inverse sine of 0.31206. So again, if we put this value into our calculator, we should find that we get anger Q to be equal to 18.1949 degrees, which to the nearest uh to the nearest 10th of a degree tells us that Q is approximately equal to 18.2 degrees. So now we have a value for Q. So we've been doing good so far. Let's put that value into our diagram. We have 18.2 degrees. And now all that's left to figure out is Anger R. What do we know that can help us? Well, again, notice that we have two angles for which we're trying to figure out the third angle and recall that in a triangle, the sum of angles equals 180 degrees. So what does that mean then? Well, by that idea, since the sum of angles in a triangle up to 180 or rather, since the sum of anger in a triangle is 180 degrees. OK. Then that tells us if we want to find angle R, all we need to do is to subtract the other two angles from 180 degrees. So that's 180 degrees minus 33.1 degrees and 18.2 degrees. Now, when we were that old, OK, we should find that we get R to be equal to 128.7 degrees. So for our triangle angle, P is 33.1 degrees. Q is 18.2 degrees and R is 128.7 degrees. B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Solve each triangle. Approximate values to the nearest tenth.

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Key Concepts

Triangle Classification and Properties

Trigonometric Ratios (Sine, Cosine, Tangent)

Law of Sines and Law of Cosines

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