To find the measure of [tex]\(\angle Q\)[/tex] using the given equation, we can follow these steps in detail:
1. Apply the Law of Cosines:
The formula given is:
[tex]\[
24^2 = 20^2 + 34^2 - 2 \cdot 20 \cdot 34 \cdot \cos(Q)
\][/tex]
Plug in the values:
[tex]\[
576 = 400 + 1156 - 1360 \cdot \cos(Q)
\][/tex]
2. Simplify the Equation:
Combine the known values on the right-hand side:
[tex]\[
576 = 1556 - 1360 \cdot \cos(Q)
\][/tex]
3. Isolate the Cosine Term:
Subtract 1556 from both sides:
[tex]\[
576 - 1556 = -1360 \cdot \cos(Q)
\][/tex]
Simplify:
[tex]\[
-980 = -1360 \cdot \cos(Q)
\][/tex]
4. Solve for [tex]\(\cos(Q)\)[/tex]:
Divide both sides by -1360:
[tex]\[
\cos(Q) = \frac{-980}{-1360}
\][/tex]
Simplify the fraction:
[tex]\[
\cos(Q) = 0.7205882352941176
\][/tex]
5. Calculate [tex]\(Q\)[/tex]:
Use the inverse cosine function (arccos) to find [tex]\(Q\)[/tex]:
[tex]\[
Q = \arccos(0.7205882352941176)
\][/tex]
In radians, this is approximately:
[tex]\[
Q \approx 0.7661460017514758 \text{ radians}
\][/tex]
6. Convert from Radians to Degrees:
Convert [tex]\(Q\)[/tex] from radians to degrees:
[tex]\[
Q \approx 43.89693239118214^\circ
\][/tex]
7. Round to the Nearest Whole Degree:
Round the angle to the nearest whole degree:
[tex]\[
Q \approx 44^\circ
\][/tex]
Thus, the measure of [tex]\(\angle Q\)[/tex] to the nearest whole degree is [tex]\(\boxed{44^\circ}\)[/tex].
