given that \\( \\cos \\theta = - \\frac { 8 } { 17 } \\) and \\( \\sin \\theta > 0 \\), determine the values…

given that \\( \\cos \\theta = - \\frac { 8 } { 17 } \\) 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 { 8 } { 17 } \\) 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 (\sin\theta)

Use the identity (\sin^{2}\theta+\cos^{2}\theta = 1). Given (\cos\theta=-\frac{8}{17}), then (\sin^{2}\theta=1-\cos^{2}\theta=1-\left(-\frac{8}{17}\right)^{2}=1-\frac{64}{289}=\frac{289 - 64}{289}=\frac{225}{289}). Since (\sin\theta>0), so (\sin\theta=\frac{15}{17}).

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{15}{17}) and (\cos\theta=-\frac{8}{17}) into the formula: (\sin2\theta=2\times\frac{15}{17}\times\left(-\frac{8}{17}\right)) (=\frac{30}{17}\times\left(-\frac{8}{17}\right)=-\frac{240}{289})

Answer:

(-\frac{240}{289})