Question

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

Explanation:

Step1: Recall sine - cosine definitions

Let the hypotenuse $c = 450$, the given angle $\theta=\frac{3\pi}{8}$, the side opposite the angle be $a$ and the side adjacent be $b$. We know that $\sin\theta=\frac{a}{c}$ and $\cos\theta=\frac{b}{c}$.

Step2: Find the side opposite the angle

Using the formula $\sin\theta=\frac{a}{c}$, we can solve for $a$. Substitute $\theta = \frac{3\pi}{8}$ and $c = 450$.
$a=c\sin\theta=450\times\sin(\frac{3\pi}{8})$.
Since $\sin(\frac{3\pi}{8})\approx0.9239$, then $a = 450\times0.9239=415.76$.

Step3: Find the side adjacent to the angle

Using the formula $\cos\theta=\frac{b}{c}$, we can solve for $b$. Substitute $\theta=\frac{3\pi}{8}$ and $c = 450$.
$b=c\cos\theta=450\times\cos(\frac{3\pi}{8})$.
Since $\cos(\frac{3\pi}{8})\approx0.3827$, then $b = 450\times0.3827 = 172.22$.

Step4: Find the other acute angle

The sum of the interior angles of a triangle is $\pi$ radians. In a right - triangle, one angle is $\frac{\pi}{2}$ radians and the given angle is $\frac{3\pi}{8}$ radians. Let the other acute angle be $\alpha$.
$\alpha=\frac{\pi}{2}-\frac{3\pi}{8}=\frac{4\pi - 3\pi}{8}=\frac{\pi}{8}$ radians.

Answer:

Length of side opposite: $415.76$
Length of side adjacent: $172.22$
Other acute angle: $\frac{\pi}{8}$ radians