Video Transcript
In the given figure, find 𝑥. Give your answer to two decimal places.
Let’s see what information we’re given in the figure. We have a triangle 𝐴𝐵𝐶. We know two of the side lengths and their included angle. And we want to know what that third side length is. All of this should lead us to think about the law of cosines. 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.
It tells us that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 times cos of 𝑐. Using this formula, the 𝑎- and 𝑏-values are the two side lengths we know and the 𝑐-value will be the included angle between those two sides. We can let lowercase 𝑎 be equal to 11, lowercase 𝑏 be equal to seven, the capital 𝐶 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 𝑥.
When we plug in what we know, we get the equation 𝑥 squared is equal to 11 squared plus seven squared minus two times 11 times seven times the cos of 46 degrees. In the first step, we can square these values and do our multiplication. From here, we get 𝑥 squared is equal to 170 minus 154 times the cos of 46 degrees. We can plug this expression into our calculator, making sure that it’s set to degrees and not to radians. When we do that, we get 63.02261 continuing.
But this is for 𝑥 squared, and we are interested in 𝑥. So, we take the square root of both sides of our equation. The square root of 𝑥 squared equals 𝑥, and the square root of 63.02261 continuing is equal to 7.93867 continuing. We’re rounding this to two decimal places. We look to the right of that, and we see an eight. So, we round our 𝑥-value up to 7.94. For this triangle, 𝑥 is equal to 7.94.
