sin(2arccos ((3)/(7)))

Use Double Angle Formula: We need to find the value of sin(2θ) where θ = arccos(3/7). We can use the double angle formula for sine, which is sin(2θ) = 2sin(θ)cos(θ).Find cos(θ): First, we find the value of cos(θ) which is given directly as cos(θ) = 3/7.Find sin(θ): To find sin(θ), we use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1. We already know cos(θ) = 3/7, so we can solve for sin(θ). sin^2(θ) = 1 - cos^2(θ) sin^2(θ) = 1 - (3/7)^2 sin^2(θ) = 1 - 9/49 sin^2(θ) = 49/49 - 9/49 sin^2(θ) = 40/49 sin(θ) = sqrt(40/49) sin(θ) = sqrt(40)/7 sin(θ) = 2sqrt(10)/7Apply Double Angle Formula: Now we can use the double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ) sin(2θ) = 2 * (2sqrt(10)/7) * (3/7) sin(2θ) = (4sqrt(10)/7) * (3/7) sin(2θ) = 12sqrt(10)/49

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Precalculus

Evaluate rational exponents

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\sin \left(2 \arccos (\frac{3}{7})\right)

Use Double Angle Formula: We need to find the value of $\sin(2\theta)$ where $\theta = \arccos(\frac{3}{7})$ . We can use the double angle formula for sine, which is $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ .
Find $\cos(\theta):$ First, we find the value of $\cos(\theta)$ which is given directly as $\cos(\theta) = \frac{3}{7}$ .
Find $\sin(\theta):$ To find $\sin(\theta)$ , we use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ . We already know $\cos(\theta) = \frac{3}{7}$ , so we can solve for $\sin(\theta)$ .
$\sin^2(\theta) = 1 - \cos^2(\theta)$
$\sin^2(\theta) = 1 - \left(\frac{3}{7}\right)^2$
$\sin^2(\theta) = 1 - \frac{9}{49}$
$\sin^2(\theta) = \frac{49}{49} - \frac{9}{49}$
$\sin^2(\theta) = \frac{40}{49}$
$\sin(\theta)$ $0$
$\sin(\theta)$ $1$
$\sin(\theta)$ $2$
Apply Double Angle Formula: Now we can use the double angle formula for sine: $\newline$ $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ $\newline$ $\sin(2\theta) = 2 \times \left(\frac{2\sqrt{10}}{7}\right) \times \left(\frac{3}{7}\right)$ $\newline$ $\sin(2\theta) = \left(\frac{4\sqrt{10}}{7}\right) \times \left(\frac{3}{7}\right)$ $\newline$ $\sin(2\theta) = \frac{12\sqrt{10}}{49}$