Question

find the value of ( x ) in the triangle shown below.
( x = square^circ )
(the triangle has sides: two sides of length 6, one side of length 5.8; one angle is ( 58^circ ), and the angle to find is ( x^circ ))

Explanation:

Step1: Identify Law of Sines

For any triangle, $\frac{\sin A}{a} = \frac{\sin B}{b}$, where $A,B$ are angles, $a,b$ are opposite sides.

Step2: Assign values to formula

Angle $58^\circ$ opposes side $5.8$; $x^\circ$ opposes side $6$.
$\frac{\sin 58^\circ}{5.8} = \frac{\sin x^\circ}{6}$

Step3: Isolate $\sin x^\circ$

Rearrange to solve for $\sin x^\circ$:
$\sin x^\circ = \frac{6 \times \sin 58^\circ}{5.8}$

Step4: Calculate $\sin x^\circ$

$\sin 58^\circ \approx 0.8480$, so:
$\sin x^\circ \approx \frac{6 \times 0.8480}{5.8} \approx \frac{5.088}{5.8} \approx 0.8772$

Step5: Solve for $x^\circ$

Take inverse sine:
$x^\circ \approx \arcsin(0.8772) \approx 61^\circ$

Answer:

$61$