Question

the angle θ is an acute angle and sin θ = 5/6. use the pythagorean identity sin²θ + cos²θ = 1 to find cos θ.
cos θ =
(simplify your answer. type an exact answer, using radicals as needed. rationalize all denominators.)

Explanation:

Step1: Substitute $\sin\theta$ into identity

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 $(\frac{5}{6})^{2}$

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

Step3: Solve for $\cos^{2}\theta$

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

Step4: Find $\cos\theta$

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

Answer:

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