### 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.