Question

use identities to find values of the sine and cosine functions of the function for the angle measure. 20. given \\( \sin \theta=-\frac{\sqrt{2}}{7} \\) and \\( \cos \theta<0 \\) \\( \cos 2 \theta= \\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression ) \\( \sin 2 \theta= \\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )

Explanation:

Step1: Find $\cos\theta$

Use the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$.
Given $\sin\theta=-\frac{\sqrt{2}}{7}$, then $\sin^{2}\theta=\frac{2}{49}$.
Substitute into the identity: $\frac{2}{49}+\cos^{2}\theta = 1$.
So, $\cos^{2}\theta=1 - \frac{2}{49}=\frac{49 - 2}{49}=\frac{47}{49}$.
Since $\cos\theta<0$, then $\cos\theta=-\frac{\sqrt{47}}{7}$.

Step2: Find $\cos2\theta$

Use the double - angle formula $\cos2\theta=1 - 2\sin^{2}\theta$.
Substitute $\sin\theta=-\frac{\sqrt{2}}{7}$ into the formula:
$\cos2\theta=1-2\times(\frac{2}{49})$.
$=1-\frac{4}{49}=\frac{49 - 4}{49}=\frac{45}{49}$.

Step3: Find $\sin2\theta$

Use the double - angle formula $\sin2\theta = 2\sin\theta\cos\theta$.
Substitute $\sin\theta=-\frac{\sqrt{2}}{7}$ and $\cos\theta=-\frac{\sqrt{47}}{7}$ into the formula:
$\sin2\theta=2\times(-\frac{\sqrt{2}}{7})\times(-\frac{\sqrt{47}}{7})$.
$=\frac{2\sqrt{94}}{49}$.

Answer:

$\cos2\theta=\frac{45}{49}$; $\sin2\theta=\frac{2\sqrt{94}}{49}$