use identities to find values of the sine and cosine functions for the angle measure. θ, given that cos 2θ =…

Explanation:

Step1: Recall the double - angle identity for cosine

The double - angle identity for cosine is $\cos2\theta = 1 - 2\sin^{2}\theta=2\cos^{2}\theta - 1$. We are given $\cos2\theta=-\frac{20}{29}$. First, use $\cos2\theta = 1 - 2\sin^{2}\theta$ to find $\sin\theta$. [ -\frac{20}{29}=1 - 2\sin^{2}\theta ] [ 2\sin^{2}\theta=1+\frac{20}{29}=\frac{29 + 20}{29}=\frac{49}{29} ] [ \sin^{2}\theta=\frac{49}{58} ] Since $0^{\circ}<\theta<90^{\circ}$, $\sin\theta>0$, so $\sin\theta=\frac{7}{\sqrt{58}}=\frac{7\sqrt{58}}{58}$.

Step2: Use the double - angle identity $\cos2\theta = 2\cos^{2}\theta - 1$ to find $\cos\theta$

[ -\frac{20}{29}=2\cos^{2}\theta - 1 ] [ 2\cos^{2}\theta=1-\frac{20}{29}=\frac{29 - 20}{29}=\frac{9}{29} ] [ \cos^{2}\theta=\frac{9}{58} ] Since $0^{\circ}<\theta<90^{\circ}$, $\cos\theta>0$, so $\cos\theta=\frac{3}{\sqrt{58}}=\frac{3\sqrt{58}}{58}$

Answer:

$\frac{3\sqrt{58}}{58}$