Question

solve the right triangle. find the length of the side opposite to the given angle. (round your answer to two decimal places.) 25.36 × find the length of the hypotenuse. (round your answer to two decimal places.) find the other acute angle.

Explanation:

Step1: Find side opposite the angle

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here $\theta = 54^{\circ}$ and adjacent side $a = 35$. So $\tan(54^{\circ})=\frac{x}{35}$, then $x = 35\times\tan(54^{\circ})\approx35\times1.376382\approx48.17$.

Step2: Find the hypotenuse

We use the cosine - function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Let the hypotenuse be $h$. So $\cos(54^{\circ})=\frac{35}{h}$, then $h=\frac{35}{\cos(54^{\circ})}\approx\frac{35}{0.587785}\approx59.55$.

Step3: Find the other acute angle

The sum of the interior angles of a triangle is $180^{\circ}$. In a right - triangle, one angle is $90^{\circ}$ and given one acute angle is $54^{\circ}$. Let the other acute angle be $y$. Then $y=180^{\circ}-90^{\circ}-54^{\circ}=36^{\circ}$.

Answer:

Length of side opposite the given angle: $48.17$
Length of the hypotenuse: $59.55$
Other acute angle: $36^{\circ}$