Question

solve the right triangle.

write your answers as integers or as decimals rounded to the nearest tenth.
ac =
ab =
m∠c = °

Explanation:

Step1: Find angle C

In a right triangle, the sum of angles is \(180^\circ\). Since \(\angle B = 90^\circ\) and \(\angle A = 34^\circ\), we have \(m\angle C=180 - 90 - 34=56^\circ\).

Step2: Find AC (hypotenuse)

We know that \(\cos A=\frac{AB}{AC}\) and \(\sin A=\frac{BC}{AC}\), but we know \(BC = 4\). Using \(\sin A=\frac{BC}{AC}\), so \(AC=\frac{BC}{\sin A}\). Since \(\sin34^\circ\approx0.5592\) and \(BC = 4\), then \(AC=\frac{4}{\sin34^\circ}\approx\frac{4}{0.5592}\approx7.2\).

Step3: Find AB

Using \(\tan A=\frac{BC}{AB}\), so \(AB=\frac{BC}{\tan A}\). \(\tan34^\circ\approx0.6745\), and \(BC = 4\), so \(AB=\frac{4}{\tan34^\circ}\approx\frac{4}{0.6745}\approx5.9\).

Answer:

\(AC\approx7.2\)
\(AB\approx5.9\)
\(m\angle C = 56^\circ\)