Given that $\sin \theta = a/b$, then $\cos \theta$ is equal to :

Question

Let $K = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ then the value of $\sin \left( 10K \frac{\pi}{3} \right)$ is :

Answer 1

To find the value of $\cos \theta$ when $\sin \theta = \frac{a}{b}$, we can utilize the Pythagorean identity:

$\sin^2 \theta + \cos^2 \theta = 1$.

Given that $\sin \theta = \frac{a}{b}$, we can substitute this into the identity:

$\left(\frac{a}{b}\right)^2 + \cos^2 \theta = 1$.

Expanding the square gives us:

$\frac{a^2}{b^2} + \cos^2 \theta = 1$.

We solve for $\cos^2 \theta$ by rearranging the equation:

$\cos^2 \theta = 1 - \frac{a^2}{b^2}$.

Simplifying the right-hand side:

$\cos^2 \theta = \frac{b^2 - a^2}{b^2}$.

To find $\cos \theta$, take the square root of both sides:

$\cos \theta = \frac{\sqrt{b^2 - a^2}}{b}$.

This confirms that the correct answer is indeed:

$\boxed{\frac{\sqrt{b^2 - a^2}}{b}}$.

Let's briefly analyze why other options are incorrect:

$\frac{b}{a}$ is incorrect because it does not relate to the value derived from the identity.
$\frac{a}{b}$ is given for $\sin \theta$, not $\cos \theta$.
$\frac{\sqrt{b^2 - a^2}}{a}$ is inconsistent with the derived formula.

This solution uses the fundamental trigonometric identities and retains consistency with the question's requirements and options.

Given that \(\sin \theta = a/b\), then \(\cos \theta\) is equal to :