Find the Reference Angle (19pi)/6

Problem

(19*π)/6

Solution

1. Identify the given angle θ=(19*π)/6 and determine its position by finding a coterminal angle within the range [0,2*π]

2. Subtract multiples of 2*π (which is (12*π)/6 from the angle to find the coterminal angle θ′

(19*π)/6−2*π=(19*π)/6−(12*π)/6

θ′=(7*π)/6

3. Determine the quadrant of the coterminal angle (7*π)/6 Since π<(7*π)/6<(3*π)/2 the angle lies in Quadrant III.

4. Apply the formula for the reference angle (θ_r) in Quadrant III, which is (θ_r)=θ′−π

(θ_r)=(7*π)/6−π

5. Simplify the expression to find the final reference angle.

(θ_r)=(7*π)/6−(6*π)/6

(θ_r)=π/6

Final Answer

(θ_r)=π/6

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