WEBVTT 00:00:00.450 --> 00:00:02.950 In the given figure, find π‘₯. 00:00:03.440 --> 00:00:06.000 Give your answer to two decimal places. 00:00:06.540 --> 00:00:09.060 Let’s see what information we’re given in the figure. 00:00:09.570 --> 00:00:11.590 We have a triangle 𝐴𝐡𝐢. 00:00:12.030 --> 00:00:16.830 We know two of the side lengths and their included angle. 00:00:17.450 --> 00:00:20.740 And we want to know what that third side length is. 00:00:21.540 --> 00:00:24.650 All of this should lead us to think about the law of cosines. 00:00:24.650 --> 00:00:35.430 The law of cosines can be used if we have two side lengths in a triangle and their included angle to find the third side length. 00:00:36.620 --> 00:00:45.980 It tells us that 𝑐 squared is equal to π‘Ž squared plus 𝑏 squared minus two π‘Žπ‘ times cos of 𝑐. 00:00:46.590 --> 00:00:58.660 Using this formula, the π‘Ž- and 𝑏-values are the two side lengths we know and the 𝑐-value will be the included angle between those two sides. 00:00:59.230 --> 00:01:16.110 We can let lowercase π‘Ž be equal to 11, lowercase 𝑏 be equal to seven, the capital 𝐢 is our angle measure, which in this case is 46 degrees, and our missing side length, lowercase 𝑐 in the formula, is going to be π‘₯. 00:01:17.100 --> 00:01:30.170 When we plug in what we know, we get the equation π‘₯ squared is equal to 11 squared plus seven squared minus two times 11 times seven times the cos of 46 degrees. 00:01:30.780 --> 00:01:34.710 In the first step, we can square these values and do our multiplication. 00:01:35.340 --> 00:01:42.590 From here, we get π‘₯ squared is equal to 170 minus 154 times the cos of 46 degrees. 00:01:43.080 --> 00:01:50.450 We can plug this expression into our calculator, making sure that it’s set to degrees and not to radians. 00:01:51.230 --> 00:01:56.240 When we do that, we get 63.02261 continuing. 00:01:56.860 --> 00:02:01.450 But this is for π‘₯ squared, and we are interested in π‘₯. 00:02:02.010 --> 00:02:05.610 So, we take the square root of both sides of our equation. 00:02:06.060 --> 00:02:19.050 The square root of π‘₯ squared equals π‘₯, and the square root of 63.02261 continuing is equal to 7.93867 continuing. 00:02:19.540 --> 00:02:22.030 We’re rounding this to two decimal places. 00:02:22.400 --> 00:02:25.380 We look to the right of that, and we see an eight. 00:02:25.920 --> 00:02:29.480 So, we round our π‘₯-value up to 7.94. 00:02:30.170 --> 00:02:34.500 For this triangle, π‘₯ is equal to 7.94.