Math Snap

STEP 1

1. The problem involves inverse trigonometric functions.
2. The identity $\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$ can be used for any $x$ in the domain of the inverse sine and cosine functions.

STEP 2

1. Apply the identity for inverse trigonometric functions.
2. Simplify the expression using the identity.
3. Calculate the final value.

STEP 3

Recall the identity for inverse trigonometric functions:
$$\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$$ This identity holds true for any $x$ such that $-1 \leq x \leq 1$.

STEP 4

Apply the identity to the given expression:
$$4 \sin^{-1}\left(\frac{\sqrt{5}}{19}\right) + 4 \cos^{-1}\left(\frac{\sqrt{5}}{19}\right)$$ Using the identity:
$$\sin^{-1}\left(\frac{\sqrt{5}}{19}\right) + \cos^{-1}\left(\frac{\sqrt{5}}{19}\right) = \frac{\pi}{2}$$ Multiply both sides of the identity by 4:
$$4 \left( \sin^{-1}\left(\frac{\sqrt{5}}{19}\right) + \cos^{-1}\left(\frac{\sqrt{5}}{19}\right) \right) = 4 \times \frac{\pi}{2}$$

Simplify the expression:
$$4 \times \frac{\pi}{2} = 2\pi$$ The value of the expression is:
$$\boxed{2\pi}$$