find the exact value of the expression, if possible. do not use a calculator.\n\\( \\cos ^ { - 1 } \\left…

find the exact value of the expression, if possible. do not use a calculator.\n\\( \\cos ^ { - 1 } \\left \\cos \\left( - \\frac { \\pi } { 3 } \\right) \\right \\)\n\\( \\bigcirc \\) a. \\( - \\frac { \\pi } { 3 } \\)\n\\( \\bigcirc \\) b. \\( \\frac { \\pi } { 3 } \\)\n\\( \\bigcirc \\) c. \\( \\frac { 2 \\pi } { 3 } \\)\n\\( \\bigcirc \\) d. \\( \\frac { 4 \\pi } { 3 } \\)

find the exact value of the expression, if possible. do not use a calculator.\n\\( \\cos ^ { - 1 } \\left \\cos \\left( - \\frac { \\pi } { 3 } \\right) \\right \\)\n\\( \\bigcirc \\) a. \\( - \\frac { \\pi } { 3 } \\)\n\\( \\bigcirc \\) b. \\( \\frac { \\pi } { 3 } \\)\n\\( \\bigcirc \\) c. \\( \\frac { 2 \\pi } { 3 } \\)\n\\( \\bigcirc \\) d. \\( \\frac { 4 \\pi } { 3 } \\)

Answer

Explanation:

Step1: Use the property of cosine function

We know that (\cos(-x)=\cos(x)), so (\cos\left(-\frac{\pi}{3}\right)=\cos\left(\frac{\pi}{3}\right)).

Step2: Recall the range of (y = \cos^{-1}(x))

The range of (y=\cos^{-1}(x)) is ([0,\pi]). For (y = \cos^{-1}(\cos\theta)), if (\theta\in[0,\pi]), then (\cos^{-1}(\cos\theta)=\theta). Since (\frac{\pi}{3}\in[0,\pi]), (\cos^{-1}\left[\cos\left(-\frac{\pi}{3}\right)\right]=\cos^{-1}\left[\cos\left(\frac{\pi}{3}\right)\right]).

Answer:

B. (\frac{\pi}{3})