Which of the following is not equivalent to sin ((pi)/(3)) ? sin ((5pi)/(3)) sin ((2pi)/(3)) sin(-(5pi)/(3)) sin ((8pi)/(3))

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Which of the following is not equivalent to  sin ((pi)/(3)) ?  sin ((5pi)/(3))  sin ((2pi)/(3))  sin(-(5pi)/(3))  sin ((8pi)/(3))

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Determine Equivalent Options: We need to determine which of the given options is not equivalent to sin((pi)/(3)). We will use the properties of the sine function, including its periodicity and symmetry, to evaluate each option.Evaluate sin((5pi)/(3)): First, let's consider sin((pi)/(3)). The sine function has a period of 2pi, which means sin(x) = sin(x + 2pi * k) for any integer k. Also, sin(x) = sin(pi - x) due to the symmetry of the sine function in the first and second quadrants.Evaluate sin((2pi)/(3)): Now, let's evaluate sin((5pi)/(3)). We can write (5pi)/(3) as (pi)/(3) + 4pi/3, which is (pi)/(3) + 2pi * (2/3). Since 2/3 is not an integer, we cannot directly apply the periodicity property. However, we can subtract 2pi from (5pi)/(3) to get a co-terminal angle: sin((5pi)/(3) - 2pi) = sin((5pi)/(3) - 6pi/3) = sin(-pi/3). Since sin(-x) = -sin(x), we have sin(-pi/3) = -sin(pi/3). Therefore, sin((5pi)/(3)) is not equivalent to sin((pi)/(3)) because it is the negative of sin((pi)/(3)).Evaluate sin(-(5pi)/(3)): Next, let's evaluate sin((2pi)/(3)). Using the symmetry property sin(x) = sin(pi - x), we have sin((2pi)/(3)) = sin(pi - (2pi)/(3)) = sin(pi/3). Therefore, sin((2pi)/(3)) is equivalent to sin((pi)/(3)).Evaluate sin((8pi)/(3)): Now, let's evaluate sin(-(5pi)/(3)). We can add 2pi to -(5pi)/(3) to find a co-terminal angle in the standard range [0, 2pi): sin(-(5pi)/(3) + 2pi) = sin(-(5pi)/(3) + 6pi/3) = sin(pi/3). Therefore, sin(-(5pi)/(3)) is equivalent to sin((pi)/(3)).Finally, let's evaluate sin((8pi)/(3)). We can subtract 2pi from (8pi)/(3) to find a co-terminal angle: sin((8pi)/(3) - 2pi) = sin((8pi)/(3) - 6pi/3) = sin((2pi)/(3)). As we determined in a previous step, sin((2pi)/(3)) is equivalent to sin((pi)/(3)). Therefore, sin((8pi)/(3)) is also equivalent to sin((pi)/(3)).

Which of the following is not equivalent to $\sin \frac{\pi}{3}$ ? $\newline$ $\sin \frac{5 \pi}{3}$ $\newline$ $\sin \frac{2 \pi}{3}$ $\newline$ $\sin \left(-\frac{5 \pi}{3}\right)$ $\newline$ $\sin \frac{8 \pi}{3}$

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