To find the measure of angle [tex]\( \angle J \)[/tex] in a triangle with sides measuring [tex]\( 11 \)[/tex], [tex]\( 13 \)[/tex], and [tex]\( 19 \)[/tex], we can use the Law of Cosines, which states:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
In this context, let's name the sides of the triangle as follows:
- [tex]\( a = 11 \)[/tex] (the side opposite angle [tex]\( J \)[/tex])
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 19 \)[/tex]
We need to find the measure of [tex]\( \angle J \)[/tex]. By rearranging the Law of Cosines formula, we have:
[tex]\[
\cos(J) = \frac{b^2 + c^2 - a^2}{2bc}
\][/tex]
Substitute the known values into the formula:
[tex]\[
\cos(J) = \frac{13^2 + 19^2 - 11^2}{2 \cdot 13 \cdot 19}
\][/tex]
Calculate the squares of the sides:
[tex]\[
13^2 = 169, \quad 19^2 = 361, \quad 11^2 = 121
\][/tex]
Substitute these values into the equation:
[tex]\[
\cos(J) = \frac{169 + 361 - 121}{2 \cdot 13 \cdot 19}
\][/tex]
Simplify the numerator:
[tex]\[
169 + 361 - 121 = 409
\][/tex]
Substitute back into the equation:
[tex]\[
\cos(J) = \frac{409}{2 \cdot 13 \cdot 19}
\][/tex]
[tex]\[
\cos(J) = \frac{409}{494}
\][/tex]
Calculate the value:
[tex]\[
\cos(J) \approx 0.82834
\][/tex]
To find [tex]\( \angle J \)[/tex], take the arccosine (inverse cosine) of this value:
[tex]\[
J \approx \arccos(0.82834)
\][/tex]
Convert this angle from radians to degrees:
[tex]\[
J \approx 34.11 \text{ degrees}
\][/tex]
When rounding to the nearest whole degree, we find:
[tex]\[
J \approx 34^\circ
\][/tex]
Thus, the measure of [tex]\( \angle J \)[/tex], rounded to the nearest whole degree, is:
[tex]\[
\boxed{34^\circ}
\][/tex]
