find the measure of the indicated angle.

Explanation:

Step1: Recall the Law of Cosines

For a triangle with sides $a, b, c$, where the angle opposite side $a$ is $\theta$, the Law of Cosines states: $$a^2 = b^2 + c^2 - 2bc\cos\theta$$ We solve for the unknown angle $\theta$ (opposite the side of length 7): $$\cos\theta = \frac{b^2 + c^2 - a^2}{2bc}$$

Step2: Substitute the given values

Here, $a=7$, $b=4$, $c=8$. Plug into the formula: $$\cos\theta = \frac{4^2 + 8^2 - 7^2}{2\times4\times8}$$

Step3: Calculate the numerator and denominator

Compute each term: $$4^2=16,\ 8^2=64,\ 7^2=49$$ $$\text{Numerator} = 16 + 64 - 49 = 31$$ $$\text{Denominator} = 2\times4\times8 = 64$$ So $\cos\theta = \frac{31}{64} \approx 0.4844$

Step4: Find the angle using arccosine

$$\theta = \arccos\left(\frac{31}{64}\right) \approx 61.0^\circ$$

Answer:

$\approx 61^\circ$