Question

the angle $\theta$ is an acute angle and $sin\theta=\frac{5}{6}$. use the pythagorean identity $sin^{2}\theta+cos^{2}\theta = 1$ to find $cos\theta$.
$cos\theta=\frac{1}{6}$
(simplify your answer. type an exact answer, using radicals as needed. rationalize all denominators.)

Explanation:

Step1: Substitute given value

Given $\sin\theta=\frac{5}{6}$, substitute into $\sin^{2}\theta+\cos^{2}\theta = 1$. So we have $(\frac{5}{6})^{2}+\cos^{2}\theta=1$.

Step2: Calculate $\sin^{2}\theta$

$(\frac{5}{6})^{2}=\frac{25}{36}$, then the equation becomes $\frac{25}{36}+\cos^{2}\theta = 1$.

Step3: Isolate $\cos^{2}\theta$

Subtract $\frac{25}{36}$ from both sides: $\cos^{2}\theta=1 - \frac{25}{36}=\frac{36 - 25}{36}=\frac{11}{36}$.

Step4: Solve for $\cos\theta$

Since $\theta$ is acute, $\cos\theta>0$. So $\cos\theta=\sqrt{\frac{11}{36}}=\frac{\sqrt{11}}{6}$.

Answer:

$\frac{\sqrt{11}}{6}$