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the number of values of x satisfying the equation sinx +sin5x=sin3x such that x€[0,pi]
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Number of Values of x Satisfying the Equation sinx sin5x = sin3x


The given equation is:

sinx sin5x = sin3x

Let's simplify this equation:

sinx (sin2x.sin3x) = sin3x

sinx (2sinx.cosx.3sinx-4sin3x) = 0

sinx(6sin^2x-4sin^3x-2cosx) = 0

sinx(2sin^2x-4sin^3x-cosx) = 0


Case 1: sinx = 0

This means that x = nπ where n is an integer.

There are a total of (π+1) values of x where sinx = 0 in the interval [0, π].


Case 2: 2sin^2x-4sin^3x-cosx = 0

Let's define a function f(x) = 2sin^2x-4sin^3x-cosx

f(x) is continuous in the interval [0, π].

f(0) = -1 and f(π) = 1.

Therefore, by the intermediate value theorem, there exists at least one value of x in the interval [0, π] where f(x) = 0.

Let's find the derivative of f(x):

f'(x) = 4sinx(3sinx-2sin^2x+cosx)

f'(x) = 4sinx(3sinx-2(1-cos^2x)+cosx)

f'(x) = 4sinx(3sinx-2+2cos^2x+cosx)

f'(x) = 4sinx(3sinx+2cos^2x+cosx-2)

f'(x) = 4sinx((3sinx+1)(sinx+2)-4)

f'(x) = 4sinx((3sinx+1)(sinx+2)-4)

Therefore, f'(x) = 0 at x = 0, π/3, and π.

f''(x) = 4(3sinx+1)(cosx)+8sinx(2sinx+cosx-3)

At x = 0, f''(0) = -4 and f''(π) = 4.

Therefore, f(x) has a local maximum at x = 0 and a local minimum at x = π.

By analyzing the sign of f'(x) and f''(x) in the different intervals, we can conclude that f(x) has:


  • Two roots in the interval [0, π
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the number of values of x satisfying the equation sinx +sin5x=sin3x such that x€[0,pi]
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