use identities to find values of the sine and cosine functions of the function for the angle measure.\n2θ…

use identities to find values of the sine and cosine functions of the function for the angle measure.\n2θ, given sinθ = \\frac{\\sqrt{2}}{5} and cosθ > 0\ncos 2θ = □\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

use identities to find values of the sine and cosine functions of the function for the angle measure.\n2θ, given sinθ = \\frac{\\sqrt{2}}{5} and cosθ > 0\ncos 2θ = □\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Find (\cos\theta) using (\sin^{2}\theta+\cos^{2}\theta = 1)

Given (\sin\theta=\frac{\sqrt{2}}{5}), then (\sin^{2}\theta=\left(\frac{\sqrt{2}}{5}\right)^{2}=\frac{2}{25}). Substitute into (\sin^{2}\theta+\cos^{2}\theta = 1): (\cos^{2}\theta=1 - \sin^{2}\theta=1-\frac{2}{25}=\frac{25 - 2}{25}=\frac{23}{25}). Since (\cos\theta>0), (\cos\theta=\frac{\sqrt{23}}{5}).

Step2: Use the double - angle formula (\cos2\theta=\cos^{2}\theta-\sin^{2}\theta)

Substitute (\sin^{2}\theta=\frac{2}{25}) and (\cos^{2}\theta=\frac{23}{25}) into the formula: (\cos2\theta=\frac{23}{25}-\frac{2}{25}) (\cos2\theta=\frac{23 - 2}{25})

Answer:

(\frac{21}{25})