Question

find the measure of the indicated angle.
49
?°
31
22
round to the nearest tenth

Explanation:

Step1: Identify the Law of Cosines

We use the Law of Cosines for a triangle with sides $a$, $b$, $c$, where the angle opposite side $c$ is $C$:
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$
Here, the side opposite the unknown angle is 22, and the other two sides are 31 and 49. Rearrange to solve for $\cos(\theta)$:
$$\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$$

Step2: Substitute the given values

Plug $a=31$, $b=49$, $c=22$ into the formula:
$$\cos(\theta) = \frac{31^2 + 49^2 - 22^2}{2 \times 31 \times 49}$$

Step3: Calculate the numerator and denominator

Compute each term:
$31^2 = 961$, $49^2 = 2401$, $22^2 = 484$
Numerator: $961 + 2401 - 484 = 2878$
Denominator: $2 \times 31 \times 49 = 3038$
$$\cos(\theta) = \frac{2878}{3038} \approx 0.9473$$

Step4: Find the inverse cosine

Calculate $\theta$ using the inverse cosine function:
$$\theta = \cos^{-1}(0.9473) \approx 18.6^\circ$$

Answer:

$18.6^\circ$