SOLUTION: please help me solve this triangle? is there more than one way using sine and cosine or just one strategy with a SAS triangle? angle B= 72 degrees, angle C = 82 degrees & si

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Question 622030: please help me solve this triangle?
is there more than one way using sine and cosine or just one strategy with a SAS triangle?
angle B= 72 degrees, angle C = 82 degrees & side b= 54
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Law of Sines
sin%28A%29%2Fa+=+sin%28B%29%2Fb+=+sin%28C%29%2Fc
Note that no matter which two fractions you use, there will be two angles and two sides in the equation.

Law of Cosines:
c%5E2+=+a%5E2%2Bb%5E2-2ab%2Acos%28C%29
or
b%5E2+=+a%5E2%2Bc%5E2-2ac%2Acos%28B%29
or
a%5E2+=+b%5E2%2Bc%5E2-2bc%2Acos%28A%29
Note that no matter which version you use, there will be 3 sides and one angle in the equation.

A basic principle in Math is that if you know all but one of the variables in an equation, you can use that equation to solve for the unknown variable. Since the Law of Sines has 4 variables, 2 angles and two (opposite) sides, you can use it when you know
  • Two angles and an opposite side; or
  • Two sides and an opposite angle.
Since the Law of Cosines has 4 variables, 3 sides and an angle, you can use it when you know
  • Three sides; or
  • Two sides and an angle
As you see, there is only one situation where you can use either the Law of Sines or Cosines: When you know two sides and an opposite angle.

Now let's look at your problem, where you know two angles and a an opposite side (which not SAS by the way). First we can easily find the third angle by using the fact that the 3 angles must add up to 180. So:
A = 180 - (72+82)
A = 26

Now we know all three angles and one side. With this information we cannot use the Law of Cosines. We must use the Law of Sines. Inserting your given values into
sin%28B%29%2Fb+=+sin%28C%29%2Fc
we get:
sin%2872%29%2F54+=+sin%2882%29%2Fc
Now we solve for c. Cross-multiplying we get:
c*sin(72) = 54*sin(82)
Dividing by sin(72) we get:
c+=+%2854%2Asin%2882%29%29%2Fsin%2872%29
Using our calculators on this we get:
c = 56.22639117
(Note: Remember, the value of any Trig function for a non-special angle will always be an approximate value. So c is approximately 56.22639117.)

Now we know two sides and three angles. With this combination of values we can use either the Law of Sines or Cosines. Normally I would pick the Law of Sines since it is a little simpler. But I'm going to use the Law of Cosines so you can that in action.

We are looking for side "a" so we will use:
a%5E2+=+b%5E2%2Bc%5E2-2bc%2Acos%28A%29
Inserting our values into this we get:
a%5E2+=+%2854%29%5E2%2B%2856.22639117%29%5E2-2%2854%29%2856.22639117%29%2Acos%2826%29
Now we simplify:
a%5E2+=+%2854%29%5E2%2B%2856.22639117%29%5E2-2%2854%29%2856.22639117%29%2A0.89879405
a%5E2+=+2916%2B3161.407064-2%2854%29%2856.22639117%29%2A0.89879405
a%5E2+=+2916%2B3161.407064-5457.88215035
a%5E2+=+619.52491365
Square root of each side (ignoring the negative square root since sides of triangles cannot be negative):
a = 24.89025740