Question

solve for x. round to the nearest tenth, if necessary.

Explanation:

Step1: Identify trig - ratio

In right - triangle $MNO$ with right - angle at $N$, we know the adjacent side to angle $M$ is $x$ and the hypotenuse is $6.3$, and $\angle M = 75^{\circ}$. We use the cosine function: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So, $\cos(75^{\circ})=\frac{x}{6.3}$.

Step2: Solve for $x$

Multiply both sides of the equation $\cos(75^{\circ})=\frac{x}{6.3}$ by $6.3$. We get $x = 6.3\times\cos(75^{\circ})$.
We know that $\cos(75^{\circ})=\cos(45^{\circ}+30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.259$.
Then $x = 6.3\times0.259 = 1.6317\approx1.6$.

Answer:

$1.6$