Question

the angle θ is an acute angle and sin θ = 7/10. use the pythagorean identity sin²θ + cos²θ = 1 to find cos θ.
cos θ = □

Explanation:

Step1: Substitute the value of $\sin\theta$ into the identity

Given $\sin\theta=\frac{7}{10}$, substitute into $\sin^{2}\theta+\cos^{2}\theta = 1$. We get $(\frac{7}{10})^{2}+\cos^{2}\theta=1$.

Step2: Calculate $(\frac{7}{10})^{2}$

$(\frac{7}{10})^{2}=\frac{49}{100}$, so the equation becomes $\frac{49}{100}+\cos^{2}\theta = 1$.

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

Subtract $\frac{49}{100}$ from both sides: $\cos^{2}\theta=1 - \frac{49}{100}=\frac{100 - 49}{100}=\frac{51}{100}$.

Step4: Find $\cos\theta$

Since $\theta$ is an acute angle, $\cos\theta>0$. So $\cos\theta=\sqrt{\frac{51}{100}}=\frac{\sqrt{51}}{10}$.

Answer:

$\frac{\sqrt{51}}{10}$