Circle 𝑚 has a radius of eight centimeters where the length 𝐶𝐵 is five centimeters. Find the angle 𝜃 in radians, giving the answer to two decimal places.
We are given in the question a sector of circle 𝑚 where the radius of the circle is equal to eight centimeters. The length of 𝐵𝑚 and 𝐶𝑚 is equal to eight centimeters. We are also told that the length of 𝐶𝐵 is five centimeters. Therefore, we have the lengths of all three sides of a triangle.
When we know all three lengths of a triangle, we can calculate any of the angles using the cosine rule or law of cosines. This states that the cosine of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all divided by two 𝑏𝑐. It is important to note that the side length we are subtracting is opposite the angle we are trying to calculate.
In this question, the cosine of angle 𝜃 is equal to eight squared plus eight squared minus five squared all divided by two multiplied by eight multiplied by eight. The right-hand side simplifies to 103 over 128. We can then take the inverse cosine of both sides of this equation so that 𝜃 is equal to the inverse cosine of 103 over 128.
Ensuring that our calculator is in radian mode, this gives us a value of 𝜃 equal to 0.6356 and so on. As we are asked to give our answer to two decimal places, 𝜃 is equal to 0.64. The angle of the sector as shown in the diagram is 0.64 radians.