Use the pattern:
\[
\sqrt{0/4}, \sqrt{1/4}, \sqrt{2/4}, \sqrt{3/4}, \sqrt{4/4}
\]
to quickly recall sine values of $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$.
Concept:
Trigonometric ratios for standard angles are derived from unit circle definitions and special right triangles. These values are fixed and commonly used in solving trigonometric equations and geometric problems.
Step 1: Evaluate $\sin(\pi/3)$
The angle $\frac{\pi}{3}$ corresponds to $60^\circ$. In a $30^\circ-60^\circ-90^\circ$ triangle, the sine of $60^\circ$ is:
\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\]
Thus, (a) matches (iii).
Step 2: Evaluate $\sin(\pi/4)$
The angle $\frac{\pi}{4}$ corresponds to $45^\circ$. In a $45^\circ-45^\circ-90^\circ$ triangle:
\[
\sin(45^\circ) = \frac{1}{\sqrt{2}}
\]
Thus, (b) matches (i).
Step 3: Evaluate $\sin(\pi/6)$
The angle $\frac{\pi}{6}$ corresponds to $30^\circ$. In a $30^\circ-60^\circ-90^\circ$ triangle:
\[
\sin(30^\circ) = \frac{1}{2}
\]
Thus, (c) matches (ii).
Step 4: Evaluate $\sin(\pi/2)$
The angle $\frac{\pi}{2}$ corresponds to $90^\circ$. On the unit circle:
\[
\sin(90^\circ) = 1
\]
Thus, (d) matches (iv).
Step 5: Final Matching
Combining all results:
\[
(a,b,c,d) = (iii, i, ii, iv)
\]
Hence, option (A) is correct.
