Question

law of sines
what is the value of z, rounded to the nearest tenth? use the law of sines to find the answer.
triangle diagram with vertices x, y, z; angle at y = 51°, angle at z = 76°, side xz = 2.6, side xy = z, side yz = x
options: 4.5 units, 2.7 units, 3.2 units
law of sines: (\frac{sin a}{a} = \frac{sin b}{b} = \frac{sin c}{c})

Explanation:

Step1: Find angle at X

The sum of angles in a triangle is \(180^\circ\). So, \(\angle X = 180^\circ - 51^\circ - 76^\circ = 53^\circ\).

Step2: Apply Law of Sines

Law of Sines: \(\frac{z}{\sin 76^\circ} = \frac{2.6}{\sin 51^\circ}\)
Solve for \(z\): \(z = \frac{2.6 \times \sin 76^\circ}{\sin 51^\circ}\)
Calculate \(\sin 76^\circ \approx 0.9703\), \(\sin 51^\circ \approx 0.7771\)
\(z = \frac{2.6 \times 0.9703}{0.7771} \approx \frac{2.5228}{0.7771} \approx 3.2\)

Answer:

3.2 units