Let's solve the given problem step by step using the law of cosines.
We have a triangle with sides \( a = 15 \), \( b = 17 \), and \( c = 8 \). We need to find the angle \(\theta\) between sides \( a \) and \( b \).
### Law of Cosines
The law of cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta) \][/tex]
Rearranging to solve for \(\cos(\theta)\):
[tex]\[ \cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Substituting the given values:
[tex]\[ \cos(\theta) = \frac{15^2 + 17^2 - 8^2}{2 \cdot 15 \cdot 17} \][/tex]
Calculating the squares:
[tex]\[ 15^2 = 225 \][/tex]
[tex]\[ 17^2 = 289 \][/tex]
[tex]\[ 8^2 = 64 \][/tex]
Substituting these into the equation:
[tex]\[ \cos(\theta) = \frac{225 + 289 - 64}{2 \cdot 15 \cdot 17} \][/tex]
[tex]\[ \cos(\theta) = \frac{450}{510} \][/tex]
[tex]\[ \cos(\theta) = 0.8823529411764706 \][/tex]
### Finding \(\theta\)
To find \(\theta\), we take the inverse cosine (arccos) of 0.8823529411764706 and convert the result to degrees:
[tex]\[ \theta = \arccos(0.8823529411764706) \][/tex]
[tex]\[ \theta \approx 28.07248693585296^{\circ} \][/tex]
### Closest Angle from Choices
We are given these angle choices:
- A. \(180^{\circ}\)
- B. \(28^{\circ}\)
- C. \(90^{\circ}\)
- D. \(62^{\circ}\)
The closest to \( 28.07248693585296^{\circ} \) is \( 28^{\circ} \).
Thus, the angle that correctly completes the law of cosines for this triangle is:
[tex]\[ \boxed{28^{\circ}} \][/tex]
