a lighthouse is located at point A. a ship travels from point B to point C. At point B,, the distance between the ship and the lighthouse is 7.5km. At point C the distance between the ship and the lighthouse is 8.6km. Angle BAC is 58 degrees. Determine the distance between B and C.
This is just a straightforward law of cosines problem. We want to find a, given A,b,c.
A=58°, c=7.5, b=8.6
a^2 = b^2+c^2 - 2bc cosA
plug and chug. . .
To find the distance between points B and C, we can use the law of cosines. The law of cosines states that for any triangle with sides a, b, and c, and an angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, side a represents the distance between the ship and the lighthouse at point B, which is given as 7.5km. Side b represents the distance between the ship and the lighthouse at point C, which is given as 8.6km. Angle BAC, which is opposite side c, is given as 58 degrees.
Let's substitute the given values into the equation and solve for c:
c^2 = 7.5^2 + 8.6^2 - 2 * 7.5 * 8.6 * cos(58)
simplifying further:
c^2 = 56.25 + 73.96 - 2 * 7.5 * 8.6 * cos(58)
Now, let's calculate the value of cos(58):
cos(58) = 0.55
Substitute this value into the equation:
c^2 = 56.25 + 73.96 - 2 * 7.5 * 8.6 * 0.55
Simplify:
c^2 = 56.25 + 73.96 - 77.55
c^2 = 52.66
Therefore, the distance between points B and C is the square root of 52.66, which is approximately 7.26 km.
Hence, the distance between B and C is approximately 7.26 km.