For x - 0- - the equation sin x-2 sin 2x -sin 3x -3 has

Sin x+2 sin 2x sin 3x =3 ⇒ sin x+4 sin x cos x 3 sin x+4 sin^3 x =3 ⇒ sin x[ 2 +4 cos x+4 (1 cos^2 x)]=3 ⇒ sin x [2 (4 cos^2 x 4 cos x+1)+1]=3 ⇒ 3 (2 cos x 1)^2 ...

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Standard XII

Mathematics

Solving Simultaneous Trigonometric Equations

For x ∈ 0, π,...

Question

For $x \in (0, π)$ , the equation sin x+2 sin 2x -sin 3x =3 has

infinitely many solutions

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three solutions

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one solution

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no solution

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Solution

The correct option is D no solution
sin x+2 sin 2x-sin 3x =3
$\Rightarrow s i n x + 4 s i n x c o s x - 3 s i n x + 4 s i n^{3} x = 3 \Rightarrow s i n x [- 2 + 4 c o s x + 4 (1 - c o s^{2} x)] = 3 \Rightarrow s i n x [2 - (4 c o s^{2} x - 4 c o s x + 1) + 1] = 3 \Rightarrow 3 - (2 c o s x - 1)^{2} = 3 c o s e c x$
Now R.H.S. $\geq 3$
But L.H.S. < 3
Hence, no solution.

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For x∈(0,π), the equation sin x+2 sin 2x -sin 3x =3 has

The correct option is D no solution sin x+2 sin 2x-sin 3x =3 ⇒sinx+4sinxcosx−3sinx+4sin3x=3⇒sinx[−2+4cosx+4(1−cos2x)]=3⇒sinx[2−(4cos2x−4cosx+1)+1]=3⇒3−(2cosx−1)2=3cosecx Now R.H.S. ≥3 But L.H.S. < 3 Hence, no solution.

For $x \in (0, π)$ , the equation sin x+2 sin 2x -sin 3x =3 has