# Question Video: Using the Cosine Rule to Find an Unknown Angle in a Triangle Mathematics

Circle π has a radius of 8 cm where the length πΆπ΅ is 5 cm. Find the angle π in radians, giving the answer to two decimal places.

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### Video Transcript

Circle π has a radius of eight centimeters where the length πΆπ΅ is five centimeters. Find the angle π in radians, giving the answer to two decimal places.

We are given in the question a sector of circle π where the radius of the circle is equal to eight centimeters. The length of π΅π and πΆπ is equal to eight centimeters. We are also told that the length of πΆπ΅ 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 π΄ is equal to π squared plus π squared minus π squared all divided by two ππ. 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 π 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 π is equal to the inverse cosine of 103 over 128.

Ensuring that our calculator is in radian mode, this gives us a value of π equal to 0.6356 and so on. As we are asked to give our answer to two decimal places, π is equal to 0.64. The angle of the sector as shown in the diagram is 0.64 radians.