given that \\( \\cos \\theta = - \\frac { 84 } { 85 } \\) and \\( \\sin \\theta > 0 \\), determine the…

Explanation:

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

Use the identity (\sin^{2}\theta+\cos^{2}\theta = 1). Given (\cos\theta=-\frac{84}{85}), then (\sin^{2}\theta=1-\cos^{2}\theta=1-\left(-\frac{84}{85}\right)^{2}=1-\frac{7056}{7225}=\frac{7225 - 7056}{7225}=\frac{169}{7225}). Since (\sin\theta>0), so (\sin\theta=\frac{13}{85}).

Step2: Calculate (\sin2\theta)

Use the double - angle formula (\sin2\theta = 2\sin\theta\cos\theta). Substitute (\sin\theta=\frac{13}{85}) and (\cos\theta=-\frac{84}{85}) into the formula: (\sin2\theta=2\times\frac{13}{85}\times\left(-\frac{84}{85}\right)=-\frac{2\times13\times84}{85\times85}=-\frac{2184}{7225}).

Step3: Calculate (\cos2\theta)

Use the double - angle formula (\cos2\theta=\cos^{2}\theta-\sin^{2}\theta). Substitute (\sin\theta=\frac{13}{85}) and (\cos\theta=-\frac{84}{85}) into the formula: (\cos2\theta=\left(-\frac{84}{85}\right)^{2}-\left(\frac{13}{85}\right)^{2}=\frac{7056}{7225}-\frac{169}{7225}=\frac{7056 - 169}{7225}=\frac{6887}{7225}).

Answer:

(\sin2\theta=-\frac{2184}{7225}) (\cos2\theta=\frac{6887}{7225})