Question

which is the best approximation of the value of \\(x\\)? use the law of sines to find the answer.

\\(\bigcirc\\) 1.5 units
\\(\bigcirc\\) 2.1 units
\\(\bigcirc\\) 2.9 units
\\(\bigcirc\\) 3.6 units

law of sines: \\(\frac{\sin(a)}{a} = \frac{\sin(b)}{b} = \frac{\sin(c)}{c}\\)

Explanation:

🆕 New Concept Discovered: Law of Sines
Relating sides and opposite angles in any triangle

Step 1: Identify the given values

In triangle \(XYZ\):

• The angle opposite to side \(x\) (which is side \(YZ\)) is \(\angle X\).
• The angle opposite to side \(XY = 2\) is \(\angle Z = 50^\circ\).
• The angle \(\angle Y = 75^\circ\).

First, find the measure of \(\angle X\) using the fact that the sum of angles in a triangle is \(180^\circ\):

$$ \angle X = 180^\circ - 75^\circ - 50^\circ = 55^\circ $$

Step 2: Set up the Law of Sines

Using the Law of Sines to relate the known side and its opposite angle to the unknown side \(x\) and its opposite angle:

$$ \frac{\sin(X)}{x} = \frac{\sin(Z)}{XY} $$

Substitute the known values into the equation:

$$ \frac{\sin(55^\circ)}{x} = \frac{\sin(50^\circ)}{2} $$

Step 3: Solve for \(x\)

Rearrange the equation to isolate \(x\):

$$ x = \frac{2 \cdot \sin(55^\circ)}{\sin(50^\circ)} $$

Calculate the trigonometric values:

• \(\sin(55^\circ) \approx 0.8192\)
• \(\sin(50^\circ) \approx 0.7660\)

Substitute these values back into the equation:

$$ x \approx \frac{2 \cdot 0.8192}{0.7660} \approx \frac{1.6384}{0.7660} \approx 2.14 $$

Rounding to the nearest tenth gives approximately \(2.1\) units.

Answer:

2.1 units