given that \\( sin \theta = \frac { 45 } { 53 } \\) and \\( cos \theta > 0 \\), determine the values of the…

given that \\( sin \theta = \frac { 45 } { 53 } \\) and \\( cos \theta > 0 \\), determine the values of the sine and cosine functions for \\( 2 \theta \\).\n\\( sin 2 \theta = \\) (type an integer or a simplified fraction.)

given that \\( sin \theta = \frac { 45 } { 53 } \\) and \\( cos \theta > 0 \\), determine the values of the sine and cosine functions for \\( 2 \theta \\).\n\\( sin 2 \theta = \\) (type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Find (\cos\theta)

Using the identity (\sin^{2}\theta+\cos^{2}\theta = 1), we have (\cos\theta=\sqrt{1-\sin^{2}\theta}). Substitute (\sin\theta=\frac{45}{53}) into the formula: (\cos\theta=\sqrt{1 - (\frac{45}{53})^{2}}=\sqrt{\frac{53^{2}-45^{2}}{53^{2}}}=\sqrt{\frac{(53 - 45)(53 + 45)}{53^{2}}}=\sqrt{\frac{8\times98}{53^{2}}}=\sqrt{\frac{784}{53^{2}}}=\frac{28}{53})

Step2: Use the double - angle formula for (\sin2\theta)

The double - angle formula for sine is (\sin2\theta=2\sin\theta\cos\theta). Substitute (\sin\theta=\frac{45}{53}) and (\cos\theta=\frac{28}{53}) into the formula: (\sin2\theta=2\times\frac{45}{53}\times\frac{28}{53}=\frac{2520}{2809})

Answer:

(\frac{2520}{2809})