given that \\( \\cos \\theta = \\frac { 40 } { 41 } \\) and \\( \\sin \\theta < 0 \\), determine the values…

given that \\( \\cos \\theta = \\frac { 40 } { 41 } \\) and \\( \\sin \\theta < 0 \\), determine the values of the sine and cosine functions for \\( 2 \\theta \\).\n\\( \\sin 2 \\theta = \\square \\) (type an integer or a simplified fraction.)

given that \\( \\cos \\theta = \\frac { 40 } { 41 } \\) and \\( \\sin \\theta < 0 \\), determine the values of the sine and cosine functions for \\( 2 \\theta \\).\n\\( \\sin 2 \\theta = \\square \\) (type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Find the value of (\sin\theta)

Use the identity (\sin^{2}\theta+\cos^{2}\theta = 1). Given (\cos\theta=\frac{40}{41}), then (\sin^{2}\theta=1-\cos^{2}\theta=1 - (\frac{40}{41})^{2}=\frac{41^{2}-40^{2}}{41^{2}}=\frac{(41 - 40)(41 + 40)}{41^{2}}=\frac{81}{41^{2}}). Since (\sin\theta<0), so (\sin\theta=-\frac{9}{41}).

Step2: Use the double - angle formula for sine

The double - angle formula for sine is (\sin2\theta = 2\sin\theta\cos\theta). Substitute (\sin\theta=-\frac{9}{41}) and (\cos\theta=\frac{40}{41}) into the formula: (\sin2\theta=2\times(-\frac{9}{41})\times\frac{40}{41}=-\frac{720}{1681}).

Answer:

(-\frac{720}{1681})