SOLUTION: A cottage under construction is to be 15.6m wide. The two sides of the roof are to be supported by equal rafters that meet at a 52 degree angle. Determine the length of the rafters

Question 1058531: A cottage under construction is to be 15.6m wide. The two sides of the roof are to be supported by equal rafters that meet at a 52 degree angle. Determine the length of the rafters to the nearest cm using
a) cosine law
b) sine law

Now, finding the answer with sine law was straightforward. Since sides b and c are equal the angles for B and C are also equal. 180 degrees - 52 degrees = 128/2 = 64 degrees for B and C.
Then use sine law to get a length of 17.8cm for either side.

Now with cosine law I am unsure. I've got three angles and a length....my textbook says I need either three lengths or two lengths and an angle in between them. I attempted anyway and this is what I got, without using the information provided by using sine law:

15.6^2 = x^2 + x^2 - 2(x)(x)cos(52)

243.36 = 2x^2 - 2x^2 cos(52)
And then I'm lost. Someone suggested these next steps:
243.36 = x^2 (2 - 2 cos(52))
x^2 = 243.36/0.769
x^2 = 316.463
x = sq root of 316.463 = 17.789 = 17.8
*Explain to me how that's legit. How did the right side of the equation turn from
2x^2 - 2x^2 cos (52)
into
x^2(2 - 2 cos(52)) ?
Thank You
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
This looks fine.
You factored out an x^2. I would have factored out a 2x^2 and had (1-cos 52) inside. Either way, you get the same answer. You had two equal lengths, so everything on the right was expressed in terms of 1 variable.
c^2=a^2+a^2-2(a)(a)cos (52) is another way to look at it.
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