Question

Find the reference angle for 16𝜋 3

Accepted Answer

Understand that the reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. Since the angle is given in radians, we will work with radians throughout. First, simplify the given angle by reducing it within one full rotation (0 to 2\pi). To do this, find the equivalent angle $$ 	heta_{reduced}  by $$subtracting multiples of $$ 2\pi $$ from $$ \frac{16\pi}{3} $$ until the angle lies between 0 and $$ 2\pi . $$Use the formula: $$ 	heta_{reduced} = 	heta - 2\pi 	imes k $$, where $$ k  is an $$integer. Calculate $$ k  by $$dividing the given angle by $$ 2\pi $$: $$ k = \left\lfloor \frac{16\pi/3}{2\pi} ightfloor . $$Then subtract $$ 2\pi k $$ from the original angle to find $$ 	heta_{reduced} . $$Determine the quadrant in which $$ 	heta_{reduced} $$ lies by comparing it to the standard quadrant boundaries: $$ 0  to$$ $$ \frac{\pi}{2}  ($$Quadrant I), $$ \frac{\pi}{2}  to$$ $$ \pi  ($$Quadrant II), $$ \pi  to$$ $$ \frac{3\pi}{2}  ($$Quadrant III), and $$ \frac{3\pi}{2}  to$$ $$ 2\pi  ($$Quadrant IV). Finally, find the reference angle $$ \alpha $$ based on the quadrant of $$ 	heta_{reduced} $$:
- Quadrant I: $$ \alpha = 	heta_{reduced} 
- $$Quadrant II: $$ \alpha = \pi - 	heta_{reduced} 
- $$Quadrant III: $$ \alpha = 	heta_{reduced} - \pi 
- $$Quadrant IV: $$ \alpha = 2\pi - 	heta_{reduced} $$

Find the reference angle for 16𝜋 3

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Key Concepts

Radian Measure

Coterminal Angles

Reference Angle