Given that $\cos \theta =\frac{15}{17}$ for an acute angle θ, calculate the value of $\sin \theta$.

The correct answer is $\frac{8}{17}$.

Step 1: Understand the given information and trigonometric identity.

We are given that $\theta$ is an acute angle and:

$\cos \theta =\frac{15}{17}$

We use the fundamental Pythagorean trigonometric identity:

${\sin }^2\theta +{\cos }^2\theta =1$

Rearranging to solve for $\sin \theta$ for an acute angle (where sine is positive):

$\sin \theta =\sqrt{1-{\cos }^2\theta }$

Step 2: Substitute the value of $\cos \theta$ into the formula.

$\sin \theta =\sqrt{1-{\left(\frac{15}{17}\right)}^2}$

Step 3: Calculate the squares and simplify.

Squaring the numerator and denominator:

${15}^2=225$

${17}^2=289$

Substitute these values back into the expression:

$\sin \theta =\sqrt{1-\frac{225}{289}}$

Convert 1 into a fraction with a common denominator of 289:

$\sin \theta =\sqrt{\frac{289-225}{289}}$

$\sin \theta =\sqrt{\frac{64}{289}}$

Step 4: Take the square root.

$\sin \theta =\frac{\sqrt{64}}{\sqrt{289}}$

$\sin \theta =\frac{8}{17}$

Thus, the value of $\sin \theta$ is $\frac{8}{17}$.