Question

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

Explanation:

Step1: Identify the known values

We know the hypotenuse $c = 2.47$ and an acute - angle $\theta=36.5^{\circ}$.

Step2: Find the adjacent side $a$

Using the cosine function $\cos\theta=\frac{a}{c}$. So $a = c\times\cos\theta$. Substituting $c = 2.47$ and $\theta = 36.5^{\circ}$, we have $a=2.47\times\cos(36.5^{\circ})\approx2.47\times0.803\approx1.98$.

Step3: Find the opposite side $b$

Using the sine function $\sin\theta=\frac{b}{c}$. So $b = c\times\sin\theta$. Substituting $c = 2.47$ and $\theta = 36.5^{\circ}$, we have $b=2.47\times\sin(36.5^{\circ})\approx2.47\times0.596\approx1.47$.

Step4: 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 one is $36.5^{\circ}$. Let the other acute - angle be $\alpha$. Then $\alpha=180^{\circ}-90^{\circ}-36.5^{\circ}=53.5^{\circ}$.

Answer:

Adjacent side $\approx1.98$, Opposite side $\approx1.47$, Other acute - angle $ = 53.5^{\circ}$