Use the Law of Cosines to find the angle C C C , rounded to the nearest tenth .

Hey, everyone. So in this practice problem, we're gonna take a look at using the law of cosines to solve for a missing angle in a non right triangle instead of a side. Let's go ahead and take a look. So we're supposed to figure out this angle c over here rounded to the nearest 10th. Now if you take a look, we can use the law of cosines or we use the law of cosines whenever you don't have a side and its opposite angle. So if we only just had these three pieces of information, we'd have to use the law of cosines. But, actually, we already have this a over here. So we could set up a law of sines, but the problem actually tells us to use a law of cosines, so let's just go ahead and do that. Alright? So if I wanna find angle c over here, then I'm gonna have to use the version of the law of cosines that includes that variable c. And remember, c always goes with c, b always goes with b, so on and so forth. So basically, the sort of version of this equation that I'm gonna use is this one over here because that involves all my variables, little a, little b, and little c, and then it involves this cosine of c over here. Okay? Let's go ahead and get started. So I've got that c squared is equal to, then I've got a squared plus b squared minus 2 ab times the cosine of c. Alright? So I'm I'm just gonna go ahead and start plugging in some numbers. C squared is 11 squared. A squared is gonna be 4 squared plus b squared, which is 9 squared minus 2 times 4 times 9 times the cosine of c. So let's just sort of tidy us tidy this up a little bit. You're always gonna have to subtract these two things to the other side. So we're gonna have to subtract, negative 4 squared and negative 9 squared. And, basically, what you're gonna end up with here is 121 minus 16 minus 81. Alright? So you got 121 minus 16 minus 81, and that's gonna equal on the on the right side, you're gonna have still this negative sign. So watch out for this negative sign. This is going to be negative 2 times 4, which is 8 times 9, which is 72. So negative 72 times the cosine of c. Alright. Remember, we're still looking for this cosine c. We just have to clean up everything else and move it off to the other side. So what I get when I subtract 1 under 21 minuteus 16 minuteus 81 is you're going to get 24. So you're going to have 24 over here and you're going to add 24, and then we're going to divide this negative 72 to the other side. So you're going to get divided by negative 72 over here. This is gonna equal your cosine of c. And if you work this out, 24 actually goes evenly into 72. So this ends up basically being 1 third negative 1 third. Alright? Now the last thing that you have to do when you're solving for an angle here is you have to take the inverse of whatever this trig function is. So So we're gonna take the inverse cosine of negative 1 thirds. And when you go ahead and plug this into your calculator, what you should get is an angle measure of a 190.5 degrees if you round it to the nearest temp. Alright? Now if you take a look at the diagram of the triangle that we've drawn here, it it actually makes a lot of sense because remember that this would be a perfectly 90 degree angle, and this angle seems to go past that. So this angle over here is bigger than 90 degrees as an obtuse angle, and that's where we got a 109.5. So this perfectly perfectly makes sense with how we've drawn our triangle. Alright? Really all there is to it. That's how you use the law of cosines to calculate an angle.

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13. Non-Right Triangles

Law of Cosines

Precalculus

13. Non-Right Triangles

Law of Cosines

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Multiple Choice

Use the Law of Cosines to find the angle $C$ , rounded to the nearest tenth.

$109.5\degree$

$50.5\degree$

$111.9\degree$

$48.1\degree$

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Verified step by step guidance

Identify the sides of the triangle: a = 4, b = 9, and c = 11. The angle opposite side c is angle C, which we need to find.

Recall the Law of Cosines formula: $ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) $. This formula relates the lengths of the sides of a triangle to the cosine of one of its angles.

Substitute the known values into the Law of Cosines formula: $ 11^2 = 4^2 + 9^2 - 2 \cdot 4 \cdot 9 \cdot \cos(C) $.

Simplify the equation: Calculate $ 11^2 $, $ 4^2 $, and $ 9^2 $, then substitute these values into the equation.

Solve for $ \cos(C) $ by isolating it on one side of the equation, and then use the inverse cosine function to find angle C. Round the result to the nearest tenth.