Select the angle that correctly completes the law of cosines for this triangle.
Check the final answer first, then review the worked steps.
Problem
Select the angle that correctly completes the law of cosines for this triangle.
Answer
28°
Step-by-step solution
- Identify the Law of Cosines: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The general form is $c^2 = a^2 + b^2 - 2ab \cos(C)$, where $C$ is the angle opposite side $c$. The problem provides an equation that is a rearrangement of this law: $8^2 + 17^2 - 2(8)(17)\cos(\_) = 15^2$. This equation is set up to find the angle opposite the side with length 15.
- Match the equation to the triangle: In the given equation, we have sides of lengths 8 and 17, and the result is $15^2$. This means that the sides $a$ and $b$ in the Law of Cosines formula are 8 and 17 (or vice versa), and the side $c$ is 15. The angle we need to find is the angle $C$ opposite the side with length 15.
- Locate the angle opposite the side of length 15: Examine the provided triangle. The side with length 15 is the base. The angle opposite this side is the angle between the sides with lengths 8 and 17. This angle is given as $28^\circ$.
- Complete the equation: Substitute the identified angle into the provided equation. The equation is $8^2 + 17^2 - 2(8)(17)\cos(\_) = 15^2$. The angle opposite the side of length 15 is $28^\circ$. Therefore, the equation becomes $8^2 + 17^2 - 2(8)(17)\cos(28^\circ) = 15^2$.