Question

Without using a calculator, determine the value of x.

Answer 1

In this triangle, we have

$\cos30^\circ=\dfrac{4x\sqrt3-5\sqrt3}{5x-4}$

You should know that $\cos30^\circ=\dfrac{\sqrt3}2$, so that

$\dfrac{\sqrt3}2=\dfrac{4x\sqrt3-5\sqrt3}{5x-4}$

Divide both sides by $\sqrt3$ to get

$\dfrac12=\dfrac{4x-5}{5x-4}$

Multiply both sides by $5x-4$ to get

$\dfrac{5x-4}2=4x-5$

Simplify and solve for $x$:

$\dfrac{5x}2-2=4x-5$

$5-2=4x-\dfrac{5x}2$

$3=\dfrac{3x}2$

$6=3x$

$\implies\boxed{x=2}$