How do you find the exact value of the third side given triangle DEF, d=sqrt3, e=5, and mangleF=pi/6?

1 Answer
Alvin L.
Dec 13, 2017

sqrt(13)~=3.61

Explanation:

Since we don't know that the triangle is a right triangle, we're required to use the Cosine Law. I will also presume that angleF is opposite the unknown side.

The Law of Cosines states that:
c^2=a^2+b^2-2abcos(C), where angleC is the angle opposite side c.

In our case, it becomes (I'll call the unknown side x):
x^2=5^2+(sqrt3)^2-10sqrt3cos(pi/6)

x^2=25+3-10sqrt3cos(pi/6)

x^2=28-10sqrt3cos(pi/6)

Next we take the square root on both sides:
x=sqrt(28-10sqrt3cos(pi/6))

cos(pi/6)=sqrt3/2

thereforex=sqrt(28-10*3/2

x=sqrt(28-30/2)=sqrt(28-15)

x=sqrt(13)

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