Final answer:
To prove that the side length of the regular hexagon inscribed in a circle of radius 6cm is 6cm, we can use sine. By applying the definition of sine and evaluating the equation, we determine that the side length of the hexagon is 5.196cm.
Step-by-step explanation:
To prove that the side length of the regular hexagon inscribed in a circle of radius 6cm is 6cm, we can use sine. In a regular hexagon, each interior angle measures 120 degrees. Since the hexagon is inscribed in a circle, the central angle that intercepts each side of the hexagon is twice the interior angle, which is 240 degrees. Using the definition of sine, where sine is equal to the ratio of the opposite side to the hypotenuse, we can let the opposite side be the side length of the hexagon and the hypotenuse be the radius of the circle.
Using the equation sin(θ) = side length / radius, we can substitute 240 degrees for θ and 6cm for the radius. Solving for the side length, we have sin(240) = x / 6cm. Taking the inverse sine of both sides, we get x = 6cm * sin(240). Evaluating sin(240) gives us x = 6cm * (-0.866) = -5.196cm. Since the side length cannot be negative, we take the positive value, and the side length of the hexagon is 5.196cm.