REFERENCE NUMBER ON THE UNIT CIRCLE

Let t be a real number. The reference number associated with t is the shortest distance along the unit circle between the terminal point determined by t and the x-axis.

The diagram below shows that to find the reference number t' it’s helpful to know the quadrant in which the terminal point determined by t lies. If the terminal point lies in quadrants I or IV, where x is positive, we find t' by moving along the circle to the positive x-axis. If it lies in quadrants II or III, where x is negative, we find t' by moving along the circle to the negative x-axis.

Finding Reference Numbers

Find the reference number for each value of t.

Example 1 :

t = 5π/6

Solution :

t' = π - 5π/6 = π/6

Example 2 :

t = 7π/4

Solution :

t' = 2π - 7π/4 = π/4

Example 3 :

t = -2π/3

Solution :

t' = π - 2π/3 = π/3

Example 4 :

t = 5.80

Solution :

t' = 2π - 5.80 ≈ 0.48

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