Question:
If \(\sin\theta\neq 0\) and
\[
\frac{1}{2}\sin\theta,\quad \cos\theta,\quad \cot\theta
\]
are in geometric progression, then the number of values of \(\theta\) lying in the interval \((-2\pi,2\pi)\) is
If \(\sin\theta\neq 0\) and
\[
\frac{1}{2}\sin\theta,\quad \cos\theta,\quad \cot\theta
\]
are in geometric progression, then the number of values of \(\theta\) lying in the interval \((-2\pi,2\pi)\) is
Show Hint
Whenever three terms are given in G.P., immediately use
\[
b^2=ac.
\]
After obtaining the trigonometric equation, solve each case separately and carefully count all solutions in the specified interval.
Updated On: Jul 9, 2026
- \(5\)
- \(7\)
- \(10\)
- \(8\) \bigskip
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
If three non-zero quantities \(a,b,c\) are in G.P., then
\[
b^2=ac.
\]
We use this condition to obtain a trigonometric equation and then count all solutions in the given interval.
Step 1: Apply the G.P. condition. Given that \[ \frac12\sin\theta,\quad \cos\theta,\quad \cot\theta \] are in G.P. Hence, \[ (\cos\theta)^2 = \left(\frac12\sin\theta\right)\cot\theta. \] Since \[ \cot\theta=\frac{\cos\theta}{\sin\theta}, \] we get \[ \cos^2\theta = \frac12\sin\theta \cdot \frac{\cos\theta}{\sin\theta}. \] Using \(\sin\theta\neq 0\), \[ \cos^2\theta = \frac12\cos\theta. \] \[ \cos\theta \left( \cos\theta-\frac12 \right) =0. \]
Step 2: Find all possible values of \(\theta\). Therefore, \[ \cos\theta=0 \] or \[ \cos\theta=\frac12. \] Case I: \[ \cos\theta=0. \] \[ \theta=\frac{\pi}{2}+n\pi. \] In the interval \((-2\pi,2\pi)\), \[ \theta= -\frac{3\pi}{2}, \; -\frac{\pi}{2}, \; \frac{\pi}{2}, \; \frac{3\pi}{2}. \] Thus, there are \[ 4 \] solutions. Case II: \[ \cos\theta=\frac12. \] \[ \theta=2n\pi\pm\frac{\pi}{3}. \] In the interval \((-2\pi,2\pi)\), \[ \theta= -\frac{5\pi}{3}, \; -\frac{\pi}{3}, \; \frac{\pi}{3}, \; \frac{5\pi}{3}. \] Thus, there are \[ 4 \] solutions.
Step 3: Count the total number of solutions. \[ 4+4=8. \]
Step 4: Write the final answer. \[ \boxed{8} \]
Step 1: Apply the G.P. condition. Given that \[ \frac12\sin\theta,\quad \cos\theta,\quad \cot\theta \] are in G.P. Hence, \[ (\cos\theta)^2 = \left(\frac12\sin\theta\right)\cot\theta. \] Since \[ \cot\theta=\frac{\cos\theta}{\sin\theta}, \] we get \[ \cos^2\theta = \frac12\sin\theta \cdot \frac{\cos\theta}{\sin\theta}. \] Using \(\sin\theta\neq 0\), \[ \cos^2\theta = \frac12\cos\theta. \] \[ \cos\theta \left( \cos\theta-\frac12 \right) =0. \]
Step 2: Find all possible values of \(\theta\). Therefore, \[ \cos\theta=0 \] or \[ \cos\theta=\frac12. \] Case I: \[ \cos\theta=0. \] \[ \theta=\frac{\pi}{2}+n\pi. \] In the interval \((-2\pi,2\pi)\), \[ \theta= -\frac{3\pi}{2}, \; -\frac{\pi}{2}, \; \frac{\pi}{2}, \; \frac{3\pi}{2}. \] Thus, there are \[ 4 \] solutions. Case II: \[ \cos\theta=\frac12. \] \[ \theta=2n\pi\pm\frac{\pi}{3}. \] In the interval \((-2\pi,2\pi)\), \[ \theta= -\frac{5\pi}{3}, \; -\frac{\pi}{3}, \; \frac{\pi}{3}, \; \frac{5\pi}{3}. \] Thus, there are \[ 4 \] solutions.
Step 3: Count the total number of solutions. \[ 4+4=8. \]
Step 4: Write the final answer. \[ \boxed{8} \]
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