Solve the triangle: b=5,c=3,A=100°b=5,c=3,A=100\degreeb=5,c=3,A=100°.

a = 6.26 , B = 51.8 ° , C = 28.2 ° a=6.26,B=51.8\degree,C=28.2\degree a = 6.26 , B = 51.8° , C = 28.2°

Solve the triangle : b = 5 , c = 3 , A = 100 ° b=5,c=3,A=100\degree b = 5 , c = 3 , A = 100° .

everyone. Welcome back. So let's take a look at how to solve this triangle. We're given 2 sides, b and c, and we're given that big a is equal to a 100 degrees. If you're not quite sure what type of triangle this is, always just look at the first step, which is sketching the triangle out. Remember, yours might look a little bit different than mine, but I'm gonna start with the fact that I know two sides and that the angle is a 100 degrees. Alright? So I know that one side is gonna be bigger than the other. So I've got a side that looks like this. This is c equals 3. And then I have an angle that's gonna be bigger than a 190 degrees. So it's gonna be an obtuse angle. It's gonna look something like this. And I know that this is gonna be side b, which equals 5. I'm gonna have to draw this a little bit longer because it's longer than c. And, therefore, what happens is when I complete this triangle, I know that this is a 100 degrees. And this totally makes sense because this is gonna be angle c. This is gonna be angle b. This is gonna be angle a, which we know is a 100 degrees. Therefore, this is side a. Alright? So what type of triangle is this? I've got 2 sides, b and c, and the angle that goes between them. So in other words, this is a side angle side triangle. I don't have a side and its opposing angle, so I can't use the law of sines. So in these types of problems where you have an SAS or SSS triangle, we're really just gonna use the law of cosines a bunch of times. Alright. Let's move on to step 2, which is we're gonna use the law of cosines. And if we have an SAS triangle like we have over here, we're gonna use step 2b, which is we're gonna find the 3rd side that we have. Alright? So in other words, in this step, we're really just gonna solve for this little a variable. I'm gonna do that over here. So in 2 a, what I'm gonna do is I'm gonna solve for a. I'm gonna have to pick the equation or the law of cosines that will give me a, and that's really just gonna be this one over here. Alright? So I'm gonna use this first sort of version of the law of cosines to solve for a, and it's pretty straightforward. This is gonna be a squared equals remember, the pattern is it's the other two variables squared, b squared and c squared, minus 2 times those two variables you just wrote, c b and c, times the cosine. And Remember, it's always gonna be the opposite of this letter over here or the capital, so it's gonna be cos. Alright? We actually know everything on the right side of this equation, so it's gonna be pretty straightforward plug and chug. What I'm gonna do is I'm gonna take these square roots, and I'm gonna start plugging a bunch of these numbers in. So that that b squared is equal to 5 squared. This This is gonna be 3 squared minus 2 times 5 times 3 times the cosine of the angle a, which is a 100 degrees. If you plug all of this into your calculator very carefully and take the square root, you should find that a is equal to 6.26. Alright? That's that last missing side that you're wanting to find out. This is 6.26. So that is step 2a. We're done with that. So now we just you have to use the law of cosines again. In step 3, what we're gonna do is we're just gonna use the law of cosines again. But now we're not solving for a side. We're going to be finding a second angle. It actually doesn't matter which angle you're trying to solve for, either big B or big C. It really doesn't matter which one you choose. What I'm gonna do is I'm just gonna go ahead and solve for big c over here just randomly. Alright? So just because I can have the option to do that. Alright. So if I wanna solve for big c, I'm gonna have to pick the equation that involves big c, which is gonna be the third version of this, law of cosines. So this is gonna be c squared equals a squared plus b squared minus 2 ab times the cosine of big c. In this step, we're gonna be do doing something a little bit different because we wanna find this, and we have everything else on the left side. So we're gonna have to isolate big c. Alright? So what I'm gonna do is I'm gonna move I'm gonna subtract the a squared and b squared over to the other side. And, basically, this just becomes 3 squared minus 6.26 squared minus 5 squared. And then on the right side, this ends up still being 2 times 6.26 times 5 times the cosine of c. Alright? Now you could solve for these things on the left and right, these numerical values. But remember, the more you round in these problems, the more prone to errors you'll make. So what I'm just gonna do here is I'm just going to sort of divide everything over here by the by itself to get rid of it and move it to the other side. And instead of plugging it into my calculator, Basically, what you're going to do is plug in your calculator this side and this side over here in parentheses. And, basically, what you should end up with is you should end up with 0.8 816. Remember, always hold on to a few more decimal places just so you can avoid those rounding errors. And this is going to equal what cosine of c is equal to. Alright? So now all we have to do is just take the inverse cosine. So if I wanna solve for c, this is just gonna be the inverse cosine of this mess over here, 0.8816. And what you should get is you should get an angle of 28.2 degrees. Alright. So that's the big answer to this big C over here. And This actually totally makes sense because if you look at your triangle, this little this big c is actually really, like, the really narrow skinny angle over here. So it makes sense that we get something really small, like 28.2 degrees. Right? So now that we've figured out what that angle is, we're gonna move on to our very last step, which is that whenever you have 2 of the angles and you're looking for the 3rd, you can always just use the angle sum formula, the fact that all the angles have to add up to a 180. So, I'm gonna just move on to my last step over here. In this last step, this is more straightforward because we already have 2 angles, a plus b plus c equals a 180 degrees. We're looking for b. So, therefore, we're just gonna subtract everything else from 180. So b is gonna equal a 180 degrees minus 100, which we know, minus the 28.2 that we found out in step 3. And if you work this out, what you're gonna get is that b is equal to 50. This is gonna be 51.8 degrees, and that's our last variable. Alright? So this this angle over here is going to be 51.8. And now we fully solve the triangle. We've got our 3 missing values. We've gotten the last side and the two angles, and and we're done. So let me know if you have any questions and I'll see you in the next one.

Solve the triangle : b = 5 , c = 3 , A = 100 ° b=5,c=3,A=100\degree b = 5 , c = 3 , A = 100° .

a = 6.26 , B = 51.8 ° , C = 28.2 ° a=6.26,B=51.8\degree,C=28.2\degree a = 6.26 , B = 51.8° , C = 28.2°

a = 6.05 , B = 50.4 ° , C = 29.6 ° a=6.05,B=50.4\degree,C=29.6\degree a = 6.05 , B = 50.4° , C = 29.6°

a = 6.26 , B = 28.2 ° , C = 51.8 ° a=6.26,B=28.2\degree,C=51.8\degree a = 6.26 , B = 28.2° , C = 51.8°

a = 6.05 , B = 29.6 ° , C = 50.4 ° a=6.05,B=29.6\degree,C=50.4\degree a = 6.05 , B = 29.6° , C = 50.4°

13. Non-Right Triangles

Law of Cosines

Precalculus

13. Non-Right Triangles

Law of Cosines

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Multiple Choice

Solve the triangle: $b=5,c=3,A=100\degree$ .

$a=6.05,B=50.4\degree,C=29.6\degree$

$a=6.26,B=28.2\degree,C=51.8\degree$

$a=6.26,B=51.8\degree,C=28.2\degree$

$a=6.05,B=29.6\degree,C=50.4\degree$

Verified step by step guidance

Identify the given values in the triangle: side b = 5, side c = 3, and angle A = 100 degrees.

Use the Law of Sines to find side a. The Law of Sines states that $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $. Start by setting up the equation $ \frac{a}{\sin 100^\circ} = \frac{5}{\sin B} $.

To find angle B, use the Law of Sines: $ \sin B = \frac{b \cdot \sin A}{a} $. Substitute the known values to solve for $ \sin B $, and then use the inverse sine function to find angle B.

Calculate angle C using the fact that the sum of angles in a triangle is 180 degrees. Use the equation $ C = 180^\circ - A - B $ to find angle C.

Finally, verify the solution by checking that the calculated side a and angles B and C satisfy the original triangle conditions and the Law of Sines.

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