Question

given that \\( \cos \theta = - \frac { 84 } { 85 } \\) and \\( \sin \theta > 0 \\), determine the values of the sine and cosine functions for \\( 2 \theta \\).
\\( \sin 2 \theta = \square \\) (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 $\sin^{2}\theta+\cos^{2}\theta = 1$.
Given $\cos\theta=-\frac{84}{85}$, then $\sin^{2}\theta=1-\cos^{2}\theta=1 - (-\frac{84}{85})^{2}=1-\frac{7056}{7225}=\frac{7225 - 7056}{7225}=\frac{169}{7225}$.
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:
$\sin2\theta=2\times\frac{13}{85}\times(-\frac{84}{85})=-\frac{2184}{7225}$.

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:
$\cos2\theta=(-\frac{84}{85})^{2}-(\frac{13}{85})^{2}=\frac{7056}{7225}-\frac{169}{7225}=\frac{7056 - 169}{7225}=\frac{6887}{7225}$.

Answer:

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