Question

25 PTS Please help :)

Answer 1

The answer is 9.968.

Answer 2

9.968

Explanation:

Law of Cosines:

`c^2=a^2+b^2 -2ab*cos(C)`

and more generally, a side squared of a triangle is equal to the other two sides squared then added, minus double the product of the other two sides and the cosine of the opposite angle, generally denoted as the capital letter of the side. (c is the side, C is the opposite angle)

in this case the other two sides are 7 and 10, so we can say that a=7, b=10 if it helps understand this a bit more, but "a" and "b" really just represent the other two sides.

the opposite angle is:
`(5\pi)/(13)` and the unknown is "a", but we can just express it as "c" for the sake of simplicity. Now plugging in all the known values we get the following equation:

`c^2=7^2+10^2-2(7)(10)cos((5\pi)/(13))\\\\c^2=49+100-140cos((5\pi)/(13))\\\\c^2=149-140cos((5\pi)/(13))`

Now from here we can just use a calculator to approximate the cosine, but make sure it's in radian mode! otherwise your answer may be incorrect as degrees are different than radians.

`c^2=149-140(0.3546)\\\\c^2=149-49.64468\\\\c^2=99.355315814\\\\c=√(99.355315814)\\\\c=9.96771367035\\\\c\approx 9.968`

So the side is approximately 9.968 units in length.