Let's work through the problem step-by-step to find the measure of angle [tex]\( Y \)[/tex] in the scalene triangle with sides 6, 11, and 12.
1. We start with the Law of Cosines formula for a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(Y)
\][/tex]
where [tex]\( c = 12 \)[/tex], [tex]\( a = 11 \)[/tex], and [tex]\( b = 6 \)[/tex].
2. Substituting the given values into the formula:
[tex]\[
12^2 = 11^2 + 6^2 - 2(11)(6) \cos(Y)
\][/tex]
3. Perform the squaring and multiplication:
[tex]\[
144 = 121 + 36 - 2(11)(6) \cos(Y)
\][/tex]
4. Simplify the equation:
[tex]\[
144 = 121 + 36 - 132 \cos(Y)
\][/tex]
5. Combine like terms:
[tex]\[
144 = 157 - 132 \cos(Y)
\][/tex]
6. Isolate the term involving [tex]\( \cos(Y) \)[/tex]:
[tex]\[
144 - 157 = -132 \cos(Y)
\][/tex]
[tex]\[
-13 = -132 \cos(Y)
\][/tex]
7. Solve for [tex]\( \cos(Y) \)[/tex]:
[tex]\[
\cos(Y) = \frac{-13}{-132} = \frac{13}{132}
\][/tex]
Simplifying the fraction, we get:
[tex]\[
\cos(Y) \approx 0.09848484848484848
\][/tex]
8. Use the inverse cosine function to find [tex]\( Y \)[/tex]:
[tex]\[
Y = \cos^{-1}(0.09848484848484848)
\][/tex]
9. Calculate the angle in radians and then convert to degrees:
[tex]\[
Y \approx 1.4721515742803193 \text{ radians}
\][/tex]
Converting to degrees:
[tex]\[
Y \approx 84.34807200980221 \text{ degrees}
\][/tex]
10. Finally, round [tex]\( Y \)[/tex] to the nearest degree:
[tex]\[
Y \approx 84 \text{ degrees}
\][/tex]
Thus, the measure of angle [tex]\( Y \)[/tex] to the nearest degree is [tex]\( \boxed{84} \)[/tex] degrees.
