Question

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

Explanation:

Step1: Find adjacent - side length

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here $\theta = 45^{\circ}$ and opposite side length $= 4$. Since $\tan45^{\circ}=1=\frac{4}{\text{adjacent}}$, the adjacent side length $a = 4$.

Step2: Find hypotenuse length

Use the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$, where $a = 4$ and $b = 4$. So $c=\sqrt{4^{2}+4^{2}}=\sqrt{16 + 16}=\sqrt{32}\approx5.66$.

Step3: Find 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 $45^{\circ}$. Let the other acute angle be $\alpha$. Then $\alpha=180^{\circ}-90^{\circ}-45^{\circ}=45^{\circ}$.

Answer:

Length of adjacent side: 4
Length of hypotenuse: 5.66
Other acute angle: $45^{\circ}$