Question

label the sides and angle and calculate the missing values in the triangle:

Explanation:

Step1: Identify known values

One - angle $\theta = 36.9^{\circ}$ and adjacent - side $x = 2.46$.

Step2: Find the opposite - side $y$

Use the tangent function $\tan\theta=\frac{y}{x}$. So $y = x\tan\theta$. Substituting $x = 2.46$ and $\theta=36.9^{\circ}$, we have $y=2.46\times\tan(36.9^{\circ})\approx2.46\times0.75 = 1.845$.

Step3: Find the hypotenuse $r$

Use the cosine function $\cos\theta=\frac{x}{r}$, then $r=\frac{x}{\cos\theta}$. Substituting $x = 2.46$ and $\theta = 36.9^{\circ}$, $\cos(36.9^{\circ})\approx0.8$, so $r=\frac{2.46}{0.8}=3.075$.

Step4: Find the other non - right angle

The sum of angles in a triangle is $180^{\circ}$. Let the other non - right angle be $\alpha$. Then $\alpha=180^{\circ}-90^{\circ}-36.9^{\circ}=53.1^{\circ}$.

Answer:

The opposite - side to the $36.9^{\circ}$ angle is approximately $1.85$, the hypotenuse is approximately $3.08$, and the other non - right angle is $53.1^{\circ}$