Choose the correct option and justify your choice : (i) 2 2 tan 30 1 tan 30 (A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30° (ii) 2 2 1 tan 45 1 tan 45 (A) tan 90° (B) 1 (C) sin 45° (D) 0 (iii) sin 2A = 2 sin A is true when A = (A) 0° (B) 30° (C) 45° (D) 60° (iv) 2 2 tan 30 1 tan 30 (A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
Step-by-Step Solution
Key Concept: Use the exact values of trigonometric ratios for 30° and 45°, the double‑angle identity \(\sin 2A = 2\sin A\cos A\), and simple algebraic manipulation. For any angle \(\theta\), \(\tan 30^{\circ}=\frac{1}{\sqrt{3}}\) and \(\tan 45^{\circ}=1\). Also \(\tan 60^{\circ}=\sqrt{3}\). The identity \(\sin 2A = 2\sin A\cos A\) reduces the equation \(\sin 2A = 2\sin A\) to \(\cos A = 1\) (or \(\sin A =0\)).
### (i) \(2\theta = 2\tan30^{\circ}+\tan30^{\circ}\)
1. \(\tan30^{\circ}=\dfrac{1}{\sqrt{3}}\).
2. \(2\tan30^{\circ}+\tan30^{\circ}=3\times\dfrac{1}{\sqrt{3}}=\sqrt{3}\).
3. \(\sqrt{3}=\tan60^{\circ}\) (since \(\tan60^{\circ}=\sqrt{3}\)).
Answer: (C) \(\tan60^{\circ}\).
### (ii) \(2\theta = 2\tan45^{\circ}-\tan45^{\circ}\)
1. \(\tan45^{\circ}=1\).
2. \(2\tan45^{\circ}-\tan45^{\circ}=2\times1-1=1\).
3. The value obtained is the number 1.
Answer: (B) 1.
### (iii) Solve \(\sin 2A = 2\sin A\)
1. Use the double‑angle identity: \(\sin 2A = 2\sin A\cos A\).
2. Set \(2\sin A\cos A = 2\sin A\).
3. Divide both sides by \(2\sin A\) (possible when \(\sin A
eq 0\)):
\[\cos A = 1\].
4. \(\cos A = 1\) gives \(A = 0^{\circ}\) (within the principal range 0°–90°).
5. If \(\sin A = 0\), then \(A = 0^{\circ}\) as well.
Answer: (A) \(0^{\circ}\).
### (iv) \(2\theta = \dfrac{2}{\tan30^{\circ}}-\dfrac{1}{\tan30^{\circ}}\)
1. \(\tan30^{\circ}=\dfrac{1}{\sqrt{3}}\).
2. Compute the reciprocals: \(\dfrac{1}{\tan30^{\circ}} = \sqrt{3}\).
3. \(\dfrac{2}{\tan30^{\circ}}-\dfrac{1}{\tan30^{\circ}} = 2\sqrt{3}-\sqrt{3}=\sqrt{3}\).
4. \(\sqrt{3}=\tan60^{\circ}\).
Answer: (C) \(\tan60^{\circ}\).
Correct Answer: {"i":"C","ii":"B","iii":"A","iv":"C"}