Question

given that \\( \cos \theta = \frac { 60 } { 61 } \\) and \\( \sin \theta < 0 \\), determine the values of the sine and cosine functions for \\( 2 \theta \\).
\\( \sin 2 \theta = - \frac { 1320 } { 3721 } \\) (type an integer or a simplified fraction.)
\\( \cos 2 \theta = \square \\) (type an integer or a simplified fraction.)

Explanation:

Step1: Find $\sin\theta$

Use the identity $\sin^{2}\theta+\cos^{2}\theta = 1$.
Since $\cos\theta=\frac{60}{61}$, then $\sin^{2}\theta=1 - (\frac{60}{61})^{2}=\frac{61^{2}-60^{2}}{61^{2}}=\frac{(61 - 60)(61 + 60)}{61^{2}}=\frac{1\times121}{61^{2}}$.
Because $\sin\theta<0$, so $\sin\theta=-\frac{11}{61}$.

Step2: Use the double - angle formula for $\cos2\theta$

The double - angle formula for cosine is $\cos2\theta=\cos^{2}\theta-\sin^{2}\theta$.
Substitute $\cos\theta=\frac{60}{61}$ and $\sin\theta =-\frac{11}{61}$ into the formula:
$\cos2\theta=(\frac{60}{61})^{2}-(-\frac{11}{61})^{2}=\frac{60^{2}-11^{2}}{61^{2}}=\frac{(60 - 11)(60 + 11)}{61^{2}}=\frac{49\times71}{61^{2}}=\frac{3479}{3721}$.

Answer:

$\cos2\theta=\frac{3479}{3721}$