Certainly! Let's find the measure of [tex]\(\angle J\)[/tex], which is the smallest angle in a triangle with sides measuring 11, 13, and 19.
The law of cosines will help us achieve this. The law of cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos A \][/tex]
In this triangle, we want to find the angle [tex]\(\angle J\)[/tex] opposite the side of length [tex]\(a = 11\)[/tex]. We will use the other two sides [tex]\(b = 13\)[/tex] and [tex]\(c = 19\)[/tex].
#### Step-by-Step Solution:
1. Substitute the known values into the law of cosines formula:
[tex]\[ 11^2 = 13^2 + 19^2 - 2 \cdot 13 \cdot 19 \cdot \cos J \][/tex]
2. Simplify the equation:
[tex]\[ 121 = 169 + 361 - 2 \cdot 13 \cdot 19 \cdot \cos J \][/tex]
[tex]\[ 121 = 530 - 494 \cos J \][/tex]
3. Isolate [tex]\(\cos J\)[/tex]:
[tex]\[ 121 - 530 = -494 \cos J \][/tex]
[tex]\[ -409 = -494 \cos J \][/tex]
4. Solve for [tex]\(\cos J\)[/tex]:
[tex]\[ \cos J = \frac{409}{494} \][/tex]
[tex]\[ \cos J \approx 0.8279 \][/tex]
5. Use the inverse cosine function to find [tex]\(\angle J\)[/tex]:
[tex]\[ J = \cos^{-1}(0.8279) \][/tex]
6. Convert the angle from radians to degrees (though typically the cosine inverse function directly gives degrees if the calculator is set to degree mode):
[tex]\[ J \approx 34.11^\circ \][/tex]
7. Round to the nearest whole degree:
[tex]\[ J \approx 34^\circ \][/tex]
Therefore, the measure of [tex]\(\angle J\)[/tex], the smallest angle in the triangle, is [tex]\(34^\circ\)[/tex].
The correct answer is [tex]\(34^\circ\)[/tex].
