Question

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

Explanation:

Step1: Recall cosine formula

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given $\theta = 75^{\circ}$ and the side opposite the right - angle (hypotenuse candidate) is considered in relation to the known adjacent side and angle. We know the side adjacent to the $75^{\circ}$ angle is found correctly as 21.44. Let the hypotenuse be $c$.

Step2: Use cosine value

$\cos75^{\circ}=\frac{21.44}{c}$. Since $\cos75^{\circ}\approx0.2588$, we can solve for $c$ by cross - multiplying: $c=\frac{21.44}{\cos75^{\circ}}$.

Step3: Calculate hypotenuse

$c=\frac{21.44}{0.2588}\approx82.85$.

Answer:

The length of the hypotenuse is approximately 82.85.