Certainly! Let's solve the problem step by step using the given information:
We are provided a triangle with side lengths:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = 6 \)[/tex]
### Applying the Law of Cosines
The law of cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \][/tex]
We want to find the measure of angle [tex]\( A \)[/tex] in this triangle. To do this, we first solve for [tex]\( \cos(A) \)[/tex]:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
### Plugging in the Values
Let's substitute the given values into the equation:
[tex]\[ \cos(A) = \frac{5^2 + 6^2 - 7^2}{2 \cdot 5 \cdot 6} \][/tex]
Calculating the squares of the sides:
[tex]\[ \cos(A) = \frac{25 + 36 - 49}{60} \][/tex]
Adding and subtracting the values in the numerator:
[tex]\[ \cos(A) = \frac{12}{60} \][/tex]
Simplifying the fraction:
[tex]\[ \cos(A) = 0.2 \][/tex]
### Finding the Angle [tex]\( A \)[/tex]
Once we have [tex]\( \cos(A) \)[/tex], we can find angle [tex]\( A \)[/tex] by taking the arccosine (inverse cosine) of 0.2. This will give us the measure of angle [tex]\( A \)[/tex] in radians first:
[tex]\[ A_{\text{radians}} = \arccos(0.2) \][/tex]
Converting this angle from radians to degrees:
[tex]\[ A_{\text{degrees}} = \text{degrees}(A_{\text{radians}}) \][/tex]
### Result
Calculating the value, we get:
- [tex]\( \cos(A) = 0.2 \)[/tex]
- [tex]\( A_{\text{radians}} \approx 1.369 \)[/tex] (in radians)
- [tex]\( A_{\text{degrees}} \approx 78.463 \)[/tex] (in degrees)
Therefore, the unknown angle [tex]\( A \)[/tex] in the given triangle, with sides [tex]\( a = 7 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = 6 \)[/tex], is approximately [tex]\( 78.463 \)[/tex] degrees.
