Question

use the pythagorean identity to find sin x. cos x = \frac{\sqrt{77}}{15} sin x = \frac{?\sqrt{}}{}}

Explanation:

Step1: Recall the Pythagorean identity

The Pythagorean identity is $\sin^{2}x+\cos^{2}x = 1$.

Step2: Substitute the given value of $\cos x$

Given $\cos x=\frac{\sqrt{77}}{15}$, then $\cos^{2}x=\frac{77}{225}$. Substituting into the identity: $\sin^{2}x+\frac{77}{225}=1$.

Step3: Solve for $\sin^{2}x$

$\sin^{2}x=1 - \frac{77}{225}=\frac{225 - 77}{225}=\frac{148}{225}$.

Step4: Simplify $\sin^{2}x$

Factor $148 = 4\times37$, so $\sin^{2}x=\frac{4\times37}{225}$.

Step5: Find $\sin x$

$\sin x=\pm\frac{2\sqrt{37}}{15}$. Since no information about the quadrant of $x$ is given, we assume the positive value (as per the form in the problem - likely expecting positive numerator).

Answer:

$\frac{2\sqrt{37}}{15}$