SATHEE: Trigonometrical Ratios and Identities 2 Question 3

Trigonometrical Ratios and Identities 2 Question 3

3. Suppose $\sin^{3} x \sin 3 x = \sum_{m = 0}^{n} C_{m} \cos n x$ is an identity in $x$ , where $C_{0}, C_{1}, \dots, C_{n}$ are constants and $C_{n} \neq 0$ . Then, the value of $n$ is… .

(1981, 2M)

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Answer:

Correct Answer: 3. 6

Solution:

Given, $\sin^{3} x \sin 3 x = \sum_{m = 0}^{n} C_{m} \cos n x$ is an identity in $x$ , where, $C_{0}, C_{1}, \dots, C_{n}$ are constants. $\sin^{3} x \sin 3 x = \frac{1}{4} 3 \sin x - \sin 3 x \cdot \sin 3 x$

$= \frac{1}{4} \frac{3}{2} \cdot 2 \sin x \cdot \sin 3 x - \sin^{2} 3 x$

$\begin{aligned} = \frac{1}{4} \frac{3}{2} (\cos 2 x - \cos x) - \frac{1}{2} (1 - \cos 6 x) \\ = \frac{1}{8} (\cos 6 x + 3 \cos 2 x - 3 \cos x - 1) \end{aligned}$

$∴$ On comparing both sides, we get $n = 6$

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Trigonometrical Ratios and Identities 2 Question 3

3. Suppose sin3⁡xsin⁡3x=∑m=0nCmcos⁡nx is an identity in x, where C0,C1,…,Cn are constants and Cn≠0. Then, the value of n is… .

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3. Suppose $\sin^{3} x \sin 3 x = \sum_{m = 0}^{n} C_{m} \cos n x$ is an identity in $x$ , where $C_{0}, C_{1}, \dots, C_{n}$ are constants and $C_{n} \neq 0$ . Then, the value of $n$ is… .