In oblique triangle ABC, C = 68°, a = 5, and b = 6. Find c to the nearest tenth.

Video transcript

Hello, today we're going to be using the law of cosines to calculate the length of Z for the given measurements. We're then going to round our answer to two decimal places. So we are told that we have a triangle XY and Z, I'm going to label the three angles as capital X, capital Y and capital Z across from angle X will be length of X across from angle Y will be the length of Y and across from angle Z will be the length of Z. Our goal is to use the use the law of cosines to calculate the length of Z. Now we are told that the angle Z is equal to degrees. We are also told that the length of X is equal to 11 ft and we are told that the length of Y is equal to 17 ft. Now, using the law of cosines, the law of cosines allows us to say state that the length of Z squared is equal to the length of X squared plus the length of Y squared minus two multiplied by the length of X multiplied by the length of Y multiplied by cosine of angle Z. So what we'll need to do is we'll need to simplify the given law of cosines. So the first thing we're going to do is we're going to plug in the values of xy and angle Z that will give us Z squared is equal to 11 squared plus 17 squared minus two, multiplied by 11 multiplied by 17, multiplied by cosine of 87 degrees. 11 squared will simplify to give us and 17 squared will simplify to give us the value of 289. Next negative two multiplied by 11 multiplied it by 17 will give us the value of negative 374. So we have negative 374 multiplied by cosine of 87 degrees. Next 121 plus 289 will give us the value of 410. So we have the length Z squared is equal to minus 374 multiplied by cosine of 87 degrees. The next thing we'll need to do is we'll need to take the square root of both sides of the equation. So we get the length of Z is equal to the square root of 410 minus 374 multiplied by cosine of 87 degrees. Now, at this point, I'd recommend plugging in this value into a calculator if you plug in the value into a calculator, you'll get a lengthy decimal of Z is equal to approximately 19.75921. Now keep in mind that the answer must be rounded to two decimal places. So if we were to round Z to two decimal places, Z will equal to 19.76 ft. This is going to be the length of Z rounded to two decimal places. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.