For x-0- then equation sin -x-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 sin^2⁡x )=3 ⇒ sin ⁡x ( 2+4 cos⁡ x+4 4 cos^2⁡x )=3 ⇒ 2+4 cos ⁡x ...

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Standard XIII

Mathematics

Solving Simultaneous Trigonometric Equations

For x∈0,π, th...

Question

For $x \in (0, π)$ , then equation $s i n x + s i n 2 x - s i n 3 x = 3$ 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

$s i n x + 2 s i n 2 x - s i n 3 x = 3 \Rightarrow s i n x + 4 s i n x c o s x - 3 s i n x + 4 s i n^{3} x = 3 \Rightarrow s i n x (- 2 + 4 c o s x + 4 s i n^{2} x) = 3 \Rightarrow s i n x (- 2 + 4 c o s x + 4 - 4 c o s^{2} x) = 3 \Rightarrow 2 + 4 c o s x - 4 c o s^{2} x = \frac{3}{s i n x} \Rightarrow 2 - (4 c o s^{2} x - 2 .2 c o s x + 1) + 1 = \frac{3}{s i n x} \Rightarrow 3 - {(2 c o s x - 1)}^{2} = \frac{3}{s i n x}$
LHS $\leq 3$ and RHS $\geq 3$
$∴$ Only possible scenario for a solution to exist is when LHS = RHS = 3. However, LHS and RHS cannot be simlultaneously equal to 3 for a particular value of $x$ . Hence no solution.

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