Which of the following is equivalent to sin ((3pi)/(7)) ? sin ((11 pi)/(7)) sin ((10 pi)/(7)) sin(-(3pi)/(7)) sin(-(11 pi)/(7))

Properties of Sine Function: Understand the properties of the sine function. The sine function has a period of 2π, which means sin(θ) = sin(θ + 2πk) for any integer k. Also, sine is an odd function, so sin(-θ) = -sin(θ).Analyze First Option: Analyze the first option sin((11pi)/(7)). We can express (11π/7) as (3π/7) + 2π(1). Since sin(θ + 2π) = sin(θ), we have sin((11pi)/(7)) = sin((3pi)/(7)).Analyze Second Option: Analyze the second option sin((10pi)/(7)). We can express (10π/7) as (3π/7) + 2π(1) - π. However, sin(θ - π) is not equal to sin(θ), so sin((10pi)/(7)) is not equivalent to sin((3pi)/(7)).Analyze Third Option: Analyze the third option sin(-(3pi)/(7)). Using the property that sine is an odd function, we have sin(-(3pi)/(7)) = -sin((3pi)/(7)). Therefore, this option is not equivalent to sin((3pi)/(7)) because it gives the negative value.Analyze Fourth Option: Analyze the fourth option sin(-(11pi)/(7)). Using the property that sine is an odd function, we have sin(-(11pi)/(7)) = -sin((11pi)/(7)). From Step 2, we know sin((11pi)/(7)) = sin((3pi)/(7)), so sin(-(11pi)/(7)) = -sin((3pi)/(7)), which is not equivalent to sin((3pi)/(7)).

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Which of the following is equivalent to
sin ((3pi)/(7)) ?

sin ((11 pi)/(7))

sin ((10 pi)/(7))

sin(-(3pi)/(7))

sin(-(11 pi)/(7))

Which of the following is equivalent to $\sin \frac{3 \pi}{7}$ ? $\newline$ $\sin \frac{11 \pi}{7}$ $\newline$ $\sin \frac{10 \pi}{7}$ $\newline$ $\sin \left(-\frac{3 \pi}{7}\right)$ $\newline$ $\sin \left(-\frac{11 \pi}{7}\right)$

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Algebra 1

Multiplication with rational exponents

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Q. Which of the following is equivalent to

\sin \frac{3 \pi}{7}

\newline

\sin \frac{11 \pi}{7}

\newline

\sin \frac{10 \pi}{7}

\newline

\sin \left(-\frac{3 \pi}{7}\right)

\newline

\sin \left(-\frac{11 \pi}{7}\right)

Properties of Sine Function: Understand the properties of the sine function. The sine function has a period of $2\pi$ , which means $\sin(\theta) = \sin(\theta + 2\pi k)$ for any integer $k$ . Also, sine is an odd function, so $\sin(-\theta) = -\sin(\theta)$ .
Analyze First Option: Analyze the first option $\sin\left(\frac{11\pi}{7}\right)$ . We can express $\frac{11\pi}{7}$ as $\frac{3\pi}{7} + 2\pi(1)$ . Since $\sin(\theta + 2\pi) = \sin(\theta)$ , we have $\sin\left(\frac{11\pi}{7}\right) = \sin\left(\frac{3\pi}{7}\right)$ .
Analyze Second Option: Analyze the second option $\sin\left(\frac{10\pi}{7}\right)$ . We can express $\frac{10\pi}{7}$ as $\frac{3\pi}{7}$ + $2$ \pi( $1$ ) - \pi. However, $\sin(\theta - \pi)$ is not equal to $\sin(\theta)$ , so $\sin\left(\frac{10\pi}{7}\right)$ is not equivalent to $\sin\left(\frac{3\pi}{7}\right)$ .
Analyze Third Option: Analyze the third option $\sin\left(-\frac{3\pi}{7}\right)$ . Using the property that sine is an odd function, we have $\sin\left(-\frac{3\pi}{7}\right) = -\sin\left(\frac{3\pi}{7}\right)$ . Therefore, this option is not equivalent to $\sin\left(\frac{3\pi}{7}\right)$ because it gives the negative value.
Analyze Fourth Option: Analyze the fourth option $\sin\left(-\frac{11\pi}{7}\right)$ . Using the property that sine is an odd function, we have $\sin\left(-\frac{11\pi}{7}\right) = -\sin\left(\frac{11\pi}{7}\right)$ . From Step $2$ , we know $\sin\left(\frac{11\pi}{7}\right) = \sin\left(\frac{3\pi}{7}\right)$ , so $\sin\left(-\frac{11\pi}{7}\right) = -\sin\left(\frac{3\pi}{7}\right)$ , which is not equivalent to $\sin\left(\frac{3\pi}{7}\right)$ .