Certainly! Let's help Keyla find the measure of angle \( Y \) step-by-step.
Given:
- Side lengths of the triangle: \( a = 11 \), \( b = 6 \), \( c = 12 \)
- Equation derived from the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(Y) \)
[tex]\[ 12^2 = 11^2 + 6^2 - 2(11)(6) \cos(Y) \][/tex]
1. Calculate \( 12^2 \):
[tex]\[ 12^2 = 144 \][/tex]
2. Calculate \( 11^2 + 6^2 \):
[tex]\[ 11^2 = 121 \][/tex]
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 121 + 36 = 157 \][/tex]
3. Substitute these values back into the Law of Cosines equation:
[tex]\[ 144 = 157 - 2(11)(6) \cos(Y) \][/tex]
4. Calculate the product \( 2(11)(6) \):
[tex]\[ 2(11)(6) = 132 \][/tex]
5. Substitute the final calculations back into the equation:
[tex]\[ 144 = 157 - 132 \cos(Y) \][/tex]
6. Solve for \( \cos(Y) \):
[tex]\[ 144 = 157 - 132 \cos(Y) \][/tex]
[tex]\[ 144 - 157 = -132 \cos(Y) \][/tex]
[tex]\[ -13 = -132 \cos(Y) \][/tex]
[tex]\[ \cos(Y) = \frac{-13}{-132} \][/tex]
[tex]\[ \cos(Y) = \frac{13}{132} \][/tex]
[tex]\[ \cos(Y) \approx 0.09848484848484848 \][/tex]
7. Use the inverse cosine function to find \( Y \):
[tex]\[ Y = \cos^{-1}(0.09848484848484848) \][/tex]
8. Convert the resulting radians to degrees:
[tex]\[ Y \approx 1.4721515742803193 \text{ radians} \][/tex]
[tex]\[ Y \approx 84^\circ \][/tex]
So, the measure of angle [tex]\( Y \)[/tex] is approximately [tex]\( \boxed{84}^\circ \)[/tex].
