SATHEE: Trigonometrical Equations 1 Question 29

Trigonometrical Equations 1 Question 29

29. Find all the solutions of $4 \cos^{2} x \sin x - 2 \sin^{2} x = 3 \sin x$ .

$(1983, 2 M)$

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Answer:

Correct Answer: 29. $x : x = n π \cup x : x = n π + (- 1)^{n} \frac{π}{10}$

U $x : x = n π + (- 1)^{n} \frac{- 3 π}{10}$

Solution:

Given, $4 \cos^{2} x \sin x - 2 \sin^{2} x = 3 \sin x$

$\Rightarrow 4 (1 - \sin^{2} x) \sin x - 2 \sin^{2} x - 3 \sin x = 0$

$\Rightarrow 4 \sin x - 4 \sin^{3} x - 2 \sin^{2} x - 3 \sin x = 0$

$\Rightarrow - 4 \sin^{3} x - 2 \sin^{2} x + \sin x = 0$

$\Rightarrow - \sin x (4 \sin^{2} x + 2 \sin x - 1) = 0$

$\Rightarrow \sin x = 0$ or $4 \sin^{2} x + 2 \sin x - 1 = 0$

$\Rightarrow \sin x = \sin 0$ or $\sin x = \frac{- 2 \pm \sqrt{4 + 16}}{2 (4)}$

$\Rightarrow x = n π$ or $\sin x = \frac{- 1 \pm \sqrt{5}}{4}$

$\Rightarrow x = n π$ or $\sin x = \sin \frac{π}{10}$

or $\sin x = \sin - \frac{3 π}{10}$

$x = n π, n π + (- 1)^{n} \frac{π}{10}, n π + (- 1)^{n} \frac{- 3 π}{10}$

$∴$ General solution set is

$\begin{aligned} x : x = n π & \cup x : x = n π + (- 1)^{n} \frac{π}{10} \\ \cup x : x = n π + (- 1)^{n} \frac{- 3 π}{10} \end{aligned}$

Trigonometrical Equations 1 Question 29

29. Find all the solutions of $4 \cos^{2} x \sin x - 2 \sin^{2} x = 3 \sin x$ .

Answer:

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Trigonometrical Equations 1 Question 29

29. Find all the solutions of 4cos2⁡xsin⁡x−2sin2⁡x=3sin⁡x.

Answer:

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29. Find all the solutions of $4 \cos^{2} x \sin x - 2 \sin^{2} x = 3 \sin x$ .