For x -0- - then equation sin x-sin 2 x-sin 3 x-3 hasA- No solutionB- One solutionC- Infinitely many solutionsD- Three solutions

The correct option is A No solution 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 ...

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Byju's Answer

Standard XII

Mathematics

Solving Simultaneous Trigonometric Equations

For x ∈0, π, ...

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.

Suggest Corrections

For x∈(0,π), then equation sin x+sin 2x−sin 3x=3 has

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