Answer :
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]
[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]
