To find the measure of an unknown angle in a triangle using the law of cosines, follow these steps. Consider a triangle with sides labeled as [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], where [tex]\( a \)[/tex] is the side opposite the angle [tex]\( A \)[/tex].
Given:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = 10 \)[/tex]
We apply the law of cosines, which states:
[tex]\[ a^2 = b^2 + c^2 - 2 b c \cos(A) \][/tex]
Substituting the given values:
[tex]\[ 5^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(A) \][/tex]
Simplify the equation:
[tex]\[ 25 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot \cos(A) \][/tex]
[tex]\[ 25 = 149 - 140 \cos(A) \][/tex]
Isolate the term involving [tex]\(\cos(A)\)[/tex]:
[tex]\[ 25 - 149 = -140 \cos(A) \][/tex]
[tex]\[ -124 = -140 \cos(A) \][/tex]
Solve for [tex]\(\cos(A)\)[/tex]:
[tex]\[ \cos(A) = \frac{-124}{-140} \][/tex]
[tex]\[ \cos(A) = \frac{124}{140} \][/tex]
[tex]\[ \cos(A) = 0.8857142857142857 \][/tex]
To find the measure of angle [tex]\( A \)[/tex], take the inverse cosine (arccos) of [tex]\( 0.8857142857142857 \)[/tex]:
[tex]\[ A = \arccos(0.8857142857142857) \][/tex]
The angle [tex]\( A \)[/tex] in degrees is:
[tex]\[ A = 27.660449899300872^\circ \][/tex]
So, the result of applying the law of cosines to this triangle is:
- [tex]\( \cos(A) = 0.8857142857142857 \)[/tex]
- [tex]\( \angle A = 27.660449899300872^\circ \)[/tex]
