MHT CET · Maths · Trigonometric Equations
The set of all possible values of \(\theta\) in the interval \((0, \pi)\) for which the points \((1,2)\) and \((\sin \theta, \cos \theta)\) lie on the same side of the line \(x+y=1\), is __________.
- A \(\left(0, \frac{\pi}{2}\right)\)
- B \(\left(0, \frac{\pi}{4}\right)\)
- C \(\left(0, \frac{3 \pi}{4}\right)\)
- D \(\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)\)
Answer & Solution
Correct Answer
(A) \(\left(0, \frac{\pi}{2}\right)\)
Step-by-step Solution
Detailed explanation
for lying on the same side \((\sin \theta+\cos \theta-1)(1+2-1)>0\)
\(\Rightarrow \sin \theta+\cos \theta>1\)
\(\Rightarrow \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta>\frac{1}{\sqrt{2}}\)
\(\Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}\)
\(\Rightarrow \frac{\pi}{4} < \theta+\frac{\pi}{4} < \frac{3 \pi}{4}\)
\(\Rightarrow 0 < \theta < \frac{\pi}{2}\)
\(\Rightarrow \sin \theta+\cos \theta>1\)
\(\Rightarrow \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta>\frac{1}{\sqrt{2}}\)
\(\Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}\)
\(\Rightarrow \frac{\pi}{4} < \theta+\frac{\pi}{4} < \frac{3 \pi}{4}\)
\(\Rightarrow 0 < \theta < \frac{\pi}{2}\)
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