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Let's work with the law of sines and cosines.

Â For example, let triangle ABC be an oblique triangle with little a = 10,

Â little b = 14 and little c = 6.

Â Let's solve the triangle.

Â Now, let's mark in the figure the information that's given.

Â We're given that little a = 10,

Â little b = 14 and little c = 6.

Â Now, this is what we refer to as the SSS case.

Â Which stands for side-side-side,

Â because we're given all three sides of the triangle, but no angles.

Â And therefore, since we are not given a matching angle-side pair,

Â we cannot start off by using the law of sines,

Â but we can however start off by using the law cosines.

Â Now, when we're in this SSS case and we're going to be using the law of cosines,

Â what we do first is we find

Â the largest angle to see if our triangle has any obtuse angles.

Â So, we'll find the largest angle first,

Â and we know what angle that is because it's across from the longest side.

Â And since little b here is the longest side,

Â that means capital B is the largest angle.

Â So, we'll be using this middle formula down here.

Â That is we have b squared which is 14 squared is equal to a squared,

Â which is 10 squared,

Â plus C squared, just 6 squared - 2,

Â times a, times c,

Â times the cosine of capital B.

Â Now, solving this equation for cosine of B,

Â it says that cosine of B = 10

Â squared + 6 squared - 14 squared,

Â all divided by 2,

Â times 10, times 6.

Â In doing this calculation,

Â we get that this is equal to negative 1/2,

Â which means that B is equal to inverse cosine of negative 1/2,

Â which is equal to 120 degrees.

Â So, let's mark that on our figure here.

Â B is 120 degrees,

Â and now we have a matching angle-side pair,

Â capital B and little b.

Â So, let's use the law of sines to find capital A.

Â That is we have sine of capital A divided by little a,

Â is equal to sine of capital B divided by little b,

Â or sine of capital A is equal to little a,

Â times sine of capital B divided by little b,

Â which is equal to 10 times sine of 120 degrees divided by 14,

Â and the sine of 120 degrees is equal to the square root of 3 over 2.

Â So, therefore this is equal to 10,

Â times the square root of 3,

Â divided by 2, still divided by fourteen,

Â which is equal to 5 square root of 3 divided by 14, and therefore,

Â A is equal to the inverse sine of 5 times the square root of 3 divided by 14,

Â and using our calculator,

Â we get that this is approximately 38.2132 degrees.

Â Now, there is another angle A that is

Â less than 180 degrees whose sine is equal to this five square root of three over 14,

Â and therefore can be a candidate for an angle in a triangle,

Â namely 180 degrees, minus this 38.2132.

Â However, that angle would be obtuse,

Â and since B is already obtuse,

Â there's no way that obtuse possibility for A would work because remember,

Â there can be at most one obtuse angle in a triangle.

Â Therefore A is this 32.2132 degrees.

Â So, let's write that in our figure up here,

Â A is approximately 38.2132 degrees.

Â Now, it still remains to find C,

Â but we can do so by using the fact that the angle measures

Â in a triangle add up to 180 degrees.

Â That is C is equal to 180 degrees minus A plus B,

Â which is approximately 180 degrees minus

Â 38.2132 degrees plus 120 degrees,

Â which is equal to 21.7868 degrees.

Â So, let's mark that on our figure as well.

Â C is approximately 21.7868 degrees.

Â Now, looking over here on the left,

Â you might be wondering why we started off by finding the largest angle first.

Â So, let's say that instead of solving for the largest angle B first,

Â we decided to solve for A first.

Â So, we'd be using this first formula down here,

Â and if we solve this for cosine of A,

Â we'd get cosine of A is equal to b squared, plus c squared,

Â minus a squared divided by two times b, times c,

Â which is equal to 14 squared plus 6

Â squared minus 10 squared divided by 2 times 14,

Â times 6, which turns out to equal 11 / 14,

Â 7:19

and therefore, A is equal to

Â inverse cosine of 11 / 14 which is approximately 38.2132,

Â which is the same answer that we found.

Â But now, if we use the law of sines to try to find capital B,

Â we have that sine of capital B divided by little b is equal to sine of capital A,

Â divided by little a or sine of B

Â is equal to b times sine of A divided by a.

Â That is sign of B is equal to

Â 14 times sine of

Â this inverse cosine of 11 divided by 14,

Â and we're using this exact value here to minimize rounding error,

Â divided by a, which is 10.

Â And therefore, B is equal to inverse sine,

Â of this whole quantity here,

Â 14 times sine of inverse cosine of 11 divided by 14,

Â all divided by 10,

Â which is equal to 60 degrees.

Â And then we'd get that C is equal to 180 degrees minus A plus

Â B which would be approximately 81.7868 degrees,

Â which is wrong, because how can capital C be larger than capital B?

Â Looking over here in our triangle,

Â little c is smaller than little b, and therefore,

Â capital C has to be smaller than capital B,

Â but what students forget,

Â looking back over here on the right,

Â this angle of 60 degrees is not

Â the only angle whose sine is equal to this quantity here, is it?

Â There's also 180 degrees minuses and looking back up here is this 120 degrees,

Â but by solving for the largest angle

Â first and assessing whether or not there's an obtuse angle in the triangle,

Â then we know that the other two angles have to be acute,

Â and we wouldn't have to worry about the other possibility for B.

Â So, if you always solve for the largest angle first,

Â then the other two have to be acute.

Â So, be careful with the SSS case.

Â Thank you and we'll see you next time.

Â