Question

the angle θ is an acute angle and sinθ = 6/7. use the pythagorean identity sin²θ + cos²θ = 1 to find cosθ. cosθ = □ (simplify your answer. type an exact answer, using radicals as needed. rationalize all denominators.)

Explanation:

Step1: Substitute sinθ into the identity

We know the Pythagorean identity $\sin^{2}\theta + \cos^{2}\theta = 1$. Given $\sin\theta=\frac{6}{7}$, we substitute it into the identity: $(\frac{6}{7})^{2}+\cos^{2}\theta = 1$.

Step2: Calculate $(\frac{6}{7})^{2}$

$(\frac{6}{7})^{2}=\frac{36}{49}$, so the equation becomes $\frac{36}{49}+\cos^{2}\theta = 1$.

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

Subtract $\frac{36}{49}$ from both sides: $\cos^{2}\theta = 1 - \frac{36}{49}$. Calculate $1 - \frac{36}{49}=\frac{49 - 36}{49}=\frac{13}{49}$.

Step4: Solve for $\cos\theta$

Take the square root of both sides. Since $\theta$ is acute, $\cos\theta$ is positive, so $\cos\theta=\sqrt{\frac{13}{49}}=\frac{\sqrt{13}}{7}$.

Answer:

$\frac{\sqrt{13}}{7}$