Question:
The solutions of $ x+\sin \,5x=\sin 3x $ in $ \left( 0,\frac{\pi }{2} \right) $ are
The solutions of $ x+\sin \,5x=\sin 3x $ in $ \left( 0,\frac{\pi }{2} \right) $ are
Updated On: Jun 23, 2024
- $ \frac{\pi }{4},\frac{\pi }{10} $
- $ \frac{\pi }{6},\frac{\pi }{3} $
- $ \frac{\pi }{4},\frac{\pi }{2} $
- $ \frac{\pi }{8},\frac{\pi }{16} $
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The Correct Option is B
Solution and Explanation
$ \sin \,\,x+\,\sin \,5x=\,\sin \,3x $
$ \Rightarrow $ $ 2\,\,\sin \,3x\,cos\,2x\,=\sin \,3x $
$ \Rightarrow $ $ \sin 3x(2\cos 2x-1)=0 $
$ \Rightarrow $ $ \sin \,\,3x=0 $
or $ 2\,\cos \,2x-1=0 $
if $ \sin \,\,3x=0 $
$ \Rightarrow $ $ 3x=0, $ or $ \pi $
$ \Rightarrow $ $ x=0 $ or $ x=\frac{\pi }{3} $
And if $ 2\,\,\cos \,2x-1=0 $
$ \Rightarrow $ $ \cos \,\,2x=\frac{1}{2}\,=\,\cos \,\frac{\pi }{3} $
$ \Rightarrow $ $ 2x=\frac{\pi }{3}\Rightarrow x=\frac{\pi }{6} $
$ \therefore $ Solutions in $ \left( 0,\frac{\pi }{2} \right) $
are $ \frac{\pi }{3},\frac{\pi }{6}. $
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.