100 followers problem

Let the triangle be $\triangle ABC$ where $\angle ABC=30^\circ$ and $\angle ACB=45^\circ$ .

Let $AD$ be the altitude drawn to $BC$ .

So,

$\begin{aligned} \tan 45^\circ & =1\Rightarrow AD=DC \\ \tan 30^\circ & = \frac{1}{\sqrt{3}} \Rightarrow \frac{AD}{\sqrt{3}+1-AD}=\frac{1}{\sqrt{3}} \\ \sqrt{3}+1-AD & =\sqrt{3} AD \\ AD=1 \\ \text{area}(\triangle ABC) & =\frac{1}{2}\cdot 1 \cdot (\sqrt{3}+1) \\ & = \boxed{1.6602} \end{aligned}$

$Circumradius=R=\dfrac{\sqrt{3}+1}{2\sin 105°}=\sqrt2~~~\small\color{#3D99F6}{(\sin105°=\frac{\sqrt3+1}{2\sqrt2})}$ Hence, Area= $\color{goldenrod}{2R^2 (\sin A)(\sin B)(\sin C)}$ $=4\times 2\times\frac{\sqrt3+1}{2\sqrt2}\times \dfrac{1}{2} \times \dfrac{1}{\sqrt2}$ $=\frac{\sqrt3+1}{2}=\huge\boxed{1.36602}$

$The ~ angle ~ opposite ~ of ~ side ~ ~ \sqrt3+1 ~ ~ is ~ 180-30-45=105. \\ \therefore ~ side ~ opposite ~ \angle ~ 30, ~ say ~ x ~ ~ ,=(\sqrt3+1)*\dfrac{Sin30}{Sin105}.\\ Area=\frac 1 2*\{\sqrt3+1\}*x*Sin45=\frac 1 2*\dfrac{(\sqrt3+1)^2}{Sin105}*Sin30*Sin45=1.366$

my solution is a bit longer, but it too yields the correct answer. What i did was that using the sine law (or whatever it is called) i marked all the angles. 30, 45, 105. sin of an angle /its opposite side =sin of the other angle/its respective opposite side. $\frac {\sin 105}{\sqrt{3} + 1}= \frac {\sin 30}{x}$ so $x = \sqrt{2}$ side opposite to 30 degree angle is $sqrt{2}$ similarly side opposite to 45 is 2. now i dropped an altitude to the $\sqrt {3} +1$ side. this becomes an isosceles triangle. so two arms containing the right angle become 1 because their hypotenuse is $sqrt{2}$ so i calculated the areas of these two new triangles which turned out to be $0.866025437844386567636 +0.5 = 1.366025437844386567636$

In general, the area of a triangle given one side length and three angles is a^2sinBsinC/(2sinA). Plugging in the values, the area is a^2sin30sin45/(2sin105) = 1.36602

In∆ABC / B=45° and / C=30°. Also BC=√3+1. By sine rule, BC sinC=AB sinA. [/_A=105°] => AB=√2. Similarly we get,AC=2. [sin105°=(√3+1)/2√2] Now,area=1/2×(sinA×AB×AC) =(√3+1)/2 =1.366