We have the following situation:
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We know the sides cc, aa and the angle between them.
Generally, when we know this, our best bet is to apply the Law of cosines, which states that, in any triangle with sides aa and cc and angle between the BB, the following relation is true:
b^2 = a^2+c^2-2ac cos Bb2=a2+c2−2accosB
This is not only true for the side bb. In fact, we have, these are equally as true:
c^2 = a^2+b^2-2abcosCc2=a2+b2−2abcosC
a^2=b^2+c^2-2bc cosAa2=b2+c2−2bccosA
Finally, let us apply this relation in our triangle:
b^2 = a^2+c^2-2ac cos Bb2=a2+c2−2accosB
b^2 = 2.35^2 +6.24^2 - 2*2.35*6.24*cos115°
In order to find cos 115°, we can write it as cos(45°+60°) and apply the Sum formula for cosine:
cos(alpha+beta) = cosalphacosbeta-sinalphasinbeta
:. cos115° = cos(45°+60^@) = sqrt2/2 * 1/2 - sqrt2/2 * sqrt3/2 =(sqrt2-sqrt6)/4
To find the exact for of b, we should rewrite a and c as the following:
b^2 = (235/100)^2 + (624/100)^2-2*235/100 * 624/100 * (sqrt2-sqrt6)/4
b^2 = 444601/10000 - 293280/10000 (sqrt2-sqrt6)/4
b^2 = (444601-73320sqrt2+73320sqrt6)/10000
color(red)(=> b=sqrt(444601-73320sqrt2+73320sqrt6)/100
