Question Video: Finding Principal Angles | Nagwa Question Video: Finding Principal Angles | Nagwa

# Question Video: Finding Principal Angles Mathematics • First Year of Secondary School

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Given the angle −(23𝜋/5), find the principal angle.

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### Video Transcript

Given the angle negative 23𝜋 over five, find the principal angle.

The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value 𝜃 radians, where 𝜃 is greater than or equal to zero and less than or equal to two 𝜋 radians. This means that we need to find the coterminal angle to negative 23𝜋 over five radians that lies between zero and two 𝜋 radians. As the given angle is negative, it will be measured in a clockwise direction. Since 23𝜋 over five is equal to four and three-fifths 𝜋 and a full rotation is equal to two 𝜋 radians, we can complete two full rotations. We then continue a further three-fifths 𝜋 radians in the clockwise direction as shown in the diagram. Since we are measuring in a clockwise direction, this angle is negative. It is negative three 𝜋 over five radians.

The principal angle must be positive. So we need to find the coterminal angle to this measured in the counterclockwise direction. To find the coterminal angle we need, and hence the principal angle, we subtract three 𝜋 over five from two 𝜋. This is equivalent to 10𝜋 over five minus three 𝜋 over five, which equals seven 𝜋 over five radians. Therefore, the principal angle of negative 23𝜋 over five is seven 𝜋 over five radians.

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