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 side P is 8.4 units, Q is 22.9 degrees and angle R is 18.1 degrees. B says it's 4.8 22.9 degrees and 18.1 degrees respectively. C 4.8 18.1 degrees and 22.9 degrees respectively. And D says it's 8.4 18.1 degrees and 22.9 degrees respectively. No, we want to figure out the missing measurements of our triangle and to make it easier to work with while we are easier to refer with, refer to while we work. Let's label each of these sides based on how we typically label triangles. So if we have angle P right here, then the side opposite to angle P would be common P. Likewise, the side opposite to anger, Q is Q and the side opposite to anger R is common R. So we know two sides and the anger between them. And we want to figure out the missing angles R and Q and the missing side. P. So how can we figure that out? Well here notice that we have two sides, two sides and an angle between them. What do we know about triangles when they have two sides and an angle between them? That is well, recall that we can apply the law of cosines here. And the law of cosines tells us that if we have two sides and an angle between them, then the square of the side opposite to the angle. In this case, P squared is going to be equal to the squares, the sum of the squares of the two other sides, Q squared plus R squared minus two multiplied by QR multiplied by the cosine of the angle between them, which in this case is P. So we can use this idea, then we can use this law to figure out the value of P. So by substituting Q squared is going to be four squared R squared. So that's gonna be five squared, that's four squared plus five squared minus two, multiplied by four, multiplied by a five, multiplied by the cosine of 139 degrees. So that is what P squared is going to be equal to. Now, I'm going to put this expression directly into the calculator just so that we can get an accurate result. Now, when you do that, OK, you should find that P squared is going to be equal to 71.18838. And if that is the value of P squared, that means P must be equal to the square root of that value. So that's the square root of 71.18838 which when we put into our calculator, we get P to be equal to 8.437 units. Notice that in our set of answers, it was rounded to the nearest 10th. So to the nearest 10th P is approximately equal to 8.4. Now that we have the value of P, so let's put that into our calculator. Oh Sorry, put that on our diagram. What else do we need? Well, no, OK, no notice that we need angle R and angle Q. Let's go back to the drawing board. So no, we know that we have three sides and we have an angle for one of the sides. We, we know the angle opposite to angle P. So no, let's ask ourselves again, what can we do to figure out any of the unknown angers? Let's take anger cue. What could we do to figure out anger cue? Well, again, notice that we have a pair of angle side and now for Q, we know the side opposite to an Q, we know that side Q is four units. So here we could use the law of science to help us figure out Anger Q because recall that by the law of science in the law of science, it tells us that the sine of an angle to its opposite side or sorry, the ratio of the sine of an angle to its opposite side is the same throughout the entire triangle. So what that means then is that the sine? OK. So let me get rid of this. It's a mark here. That means the sine of anger Q two, the side Q is going to be the same ratio as the S of anger P to side P which is going to be the same as the sine of anger R to side R. Now, as we said in our problem, we know the sign of, we know what anger P is and the length of P. So if we can figure what this ratio is, we can use it to figure out the value of Q. So applying that idea, then that means the sine of anger Q that we want to its opposite side, which is four units is going to be equal to the sign of anger P which is 139 degrees to the length of P which we just found to be 8.437 units. Now to solve for the sine of Q, we can multiply both sides by four. So the sine of Q equals four multiplied by the sine of 139 degrees. OK? Divided by 8.437. OK? This expression when we put it into our calculator OK, we get the sine of Q to be equal to 0.31104. OK? And now if we're going to figure out what the value of Q is, then we can find the inverse of that expression. In other words, Q is going to be equal to the inverse sign of 0.31104 which is equal. Oops my apologies, which is going to be equal to 18.122 degrees or approximately 18.1 degrees. So now we have a value for a Q. So now that we know that Q, let me write it inside. Now that we know that Q is 18.1 degrees, we have the value of side P. Finally, we just need to figure out an R. Now here, the thing is four R, OK. Notice that we have two angles in the triangle and record that angles in a triangle come to 180 degrees. OK. So since angles in a triangle angles in a triangle, some to 180 degrees, that means angle R is going to be equal to 180 degrees minus the sum of the other two angles, that's 100 and 39 degrees and 18.1 degrees. I know when we do that, when we put this into our calculator again, when we figure that out, this tells us that angle R is 22.9 degrees. So for our triangle. The missing side is approximately 8.4 units. Angle Q is 18.1 degrees and angle R is 22.9 degrees. If we look back on our answer choices, that means answer choice. D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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