Question

Match List-I with List-II. 

• A. a - iii, b - i, c - ii, d - iv
• B. a - iv, b - ii, c - i, d - iii
• C. a - iii, b - ii, c - i, d - iv
• D. a - iv, b - ii, c - iii, d - i

Hint:

The four most important sine values are \[ \sin 0^\circ=0,\quad \sin 30^\circ=\frac{1}{2},\quad \sin 45^\circ=\frac{1}{\sqrt{2}},\quad \sin 60^\circ=\frac{\sqrt{3}}{2},\quad \sin 90^\circ=1. \] These values appear frequently in competitive examinations.

Answer 1

The Correct Option is A

Concept: Certain trigonometric values are standard and must be memorized. \[ \sin 30^\circ=\frac{1}{2} \] \[ \sin 45^\circ=\frac{1}{\sqrt{2}} \] \[ \sin 60^\circ=\frac{\sqrt{3}}{2} \] \[ \sin 90^\circ=1. \] Since \[ 30^\circ=\frac{\pi}{6}, \quad 45^\circ=\frac{\pi}{4}, \quad 60^\circ=\frac{\pi}{3}, \quad 90^\circ=\frac{\pi}{2}, \] these values can be used directly.

Step 1: Evaluate \(\sin(\pi/3)\). \[ \sin\left(\frac{\pi}{3}\right) = \sin 60^\circ = \frac{\sqrt{3}}{2}. \] Thus, \[ (a)\rightarrow(iii). \]

Step 2: Evaluate \(\sin(\pi/4)\). \[ \sin\left(\frac{\pi}{4}\right) = \sin 45^\circ = \frac{1}{\sqrt{2}}. \] Thus, \[ (b)\rightarrow(i). \]

Step 3: Evaluate \(\sin(\pi/6)\). \[ \sin\left(\frac{\pi}{6}\right) = \sin 30^\circ = \frac{1}{2}. \] Thus, \[ (c)\rightarrow(ii). \]

Step 4: Evaluate \(\sin(\pi/2)\). \[ \sin\left(\frac{\pi}{2}\right) = \sin 90^\circ = 1. \] Thus, \[ (d)\rightarrow(iv). \] Conclusion: The correct matching is \[ a-iii,\quad b-i,\quad c-ii,\quad d-iv. \] Hence the correct option is \[ \boxed{(A)}. \]