Solve each triangle. See Examples 2 and 3.

A ...

Question

Solve each triangle. See Examples 2 and 3.

A = 80° 40', b = 143 cm, c = 89.6 cm

Accepted Answer

Welcome back. Everyone in this problem. We want to find the missing measurements of the triangle where angle P is 70 degrees 20 minutes. Side Q is 109 ft long and side R is 92.3 ft long. For our answer choices. A says side P is 117 ft. Angle Q is 61 degrees 33 minutes and angle R is 48 degrees seven minutes. B says P is 124 ft. Angle Q is 61 degrees 33 minutes and R angle R is 48 degrees seven minutes. C says P is 117 ft. Angle Q is 48 degrees seven minutes and angle R is 61 degrees 33 minutes. And the D says side P is 124 ft. Angle Q is 48 degrees seven minutes and angle R is 61 degrees 33 minutes. Now, we're going to find the missing measurements of this triangle. OK. It would make sense for us to try and sketch what's going on here. Remember a sketch doesn't have to be exact, but it's just to give us an idea. Of where these missing measurements may be. Now for our triangle here is Labour triangle P. QR K where if angle P is right here, the side opposite to angle P is side P denoted in as a command P. Likewise, Anger Q, the side opposite to anger Q would be side Q and the side opposite to anger R would be side R. And we know that side R is 92.3 ft. Side Q is oops, let me fix that side queue is 109 ft and angle P is 70 degrees 20 minutes. Let me put that here. So this is all of what we know so far. Now, let's think about it if we notice we have two sides and the angle between them. OK? Let's highlight that there. And for those two sides and the angle between them, we want to know the side that's opposite to the angle. So the question is, what do we know that relates two sides and the anger between them to the side opposite to the anger? Well, we can think the law of core science recall that by the law of core science, OK? By the law of course, science, we know that P squared, OK is going to be equal to R squared plus Q squared. Let me write it Q squared plus R squared. OK? The, the other two sides minus two multiplied by Q multiplied by R multiplied by the cosine of P. So if we, we, we know all of these values, we know a side Q, side R and the P. So we should be able to solve for P squared because no, that means P squared, OK, is going to be equal to 109 squared plus 92.3 square eh minus two, multiplied by 109 by about 92.3 multiplied by the cosine of P core sine of 70 degrees and 20 minutes. Now, we can put this expression in our calculator. OK. So let's try to figure out what that is. When you put that into your calculator, you should find that P squared, OK is approximately equal to 314,628 0.37338 square units depending on the calculator that you're using. This tells us then that to solve for P, we can find the square root of both sides. So P is the square root of that value which would be equal to 117 to the nearest foot. So P is approximately equal to 117 ft. So now we have a value for the length of side P. OK. Let's put that on our diagram. So now we know what P is equal to, let me clear what we have here. Next, we need to figure out our unknown angles Q and R. So how can we figure that out? Well, now that we've found the value of side P, we have a pair of angles and sides. OK? We have a pair of anger and its opposite side. And no, if we want to figure out any of the other angles, let's say that we wanted to figure out angle P or no. Rather, let's say we want to figure out angle R because we know angle P already notice that we know the value of the side opposite to anger R. So the question is what relates or yeah, what relates a pair of known anger and side to another anger that we don't know but we know its side. Well, we can think the law of science recall that the law of science tells us that the ratio of the sine of an angle to its opposite side is the same through the triangle. So if we know a pair of angle on the opposite side, we can use that idea to figure out the other angle once we can find the ratio. So let's use the law of science to solve for R. Now by the law of science. OK? It tells us then that the ratio of the sine of R to side R is going to be equal to the sine of an P to side P. Now we know that side R, you know the length of side R, let me fix that right here. OK? We know the length of side R is uh 92.3 ft. OK. We know that angle P is 70 degrees and 20 minutes. And we also know we just found out that the length of side P is 117 ft. That means then that the sign of R is going to be equal to 92.3 multiplied by the sine of 70 degrees and 20 minutes divided by 117. That is, we multiplied both sides by 92.3. And know if the sine of an angle equals an expression, then we can find the angle by using the inverse sine. In other words, R is going to be equal to the inverse sine of 92.3 cyn 70 degrees, 20 minutes divided by 117. So to solve for the value of R, we can put this expression into our calculator. Now, when you put this into your calculator, what do you find? Well, we should find that angle R is approximately equal to 48 degrees and seven minutes to the nearest minute. So now we have a value for an R. Let's put that value for an art onto our sketch. So now we know that I to get rid of these here. Oops. Now we know that anger R is 48 degrees and seven minutes. All we have left to find is the value of Anger Q. Now, if we look at the diagram here, we know that we have two angles. And we want to find the third angle in the triangle. Let's think, do we know anything that relates the angles in a triangle? Well, we know that the sum of angles in a triangle is 180 degrees. So P plus Q plus R equals 180. Therefore, we can find Q by subtracting angles P and R from 180 degrees. So by that idea, let's move to the, let's move to the left to the right here. OK? Buy the Angus some property of triangles. Then that means anger cue. Anger cue is going to be equal to 180 degrees minus angle P which is 70 degrees, 20 minutes minus angle R which is 48 degrees seven minutes. Now, when we work that out, we find that angle Q equals 61 degrees and 33 minutes. Therefore, side P is 117 ft. Angle R angle Q is 61 degrees and 33 minutes and angle R is 48 degrees and seven minutes. If we look back on our answer choices, that would have been answer choice. A thanks a lot for watching everyone. I hope this video helps.

Solve each triangle. See Examples 2 and 3.

A = 80° 40', b = 143 cm, c = 89.6 cm

Key Concepts

Law of Sines

Angle Conversion

Triangle Properties

Watch next