find the reference angle for $\\frac{4\\pi}{3}$. the reference angle is $\\square$. (simplify your answer…

find the reference angle for $\\frac{4\\pi}{3}$. the reference angle is $\\square$. (simplify your answer. use integers or fractions for any numbers in the expression. type your answer in radians.)

find the reference angle for $\\frac{4\\pi}{3}$. the reference angle is $\\square$. (simplify your answer. use integers or fractions for any numbers in the expression. type your answer in radians.)

Answer

Explanation:

Step1: Determine the quadrant

The angle (\theta=\frac{4\pi}{3}). Since (\pi<\frac{4\pi}{3}<\frac{3\pi}{2}), the angle (\frac{4\pi}{3}) lies in the third quadrant.

Step2: Use the formula for reference angle in the third quadrant

For an angle (\theta) in the third quadrant, the reference angle (\theta') is given by (\theta'=\theta - \pi). Substitute (\theta=\frac{4\pi}{3}) into the formula: (\theta'=\frac{4\pi}{3}-\pi). Since (\pi = \frac{3\pi}{3}), then (\theta'=\frac{4\pi}{3}-\frac{3\pi}{3}).

Step3: Simplify the expression

(\theta'=\frac{4\pi - 3\pi}{3}=\frac{\pi}{3}).

Answer:

(\frac{\pi}{3})