$f (x) = \sqrt{{(cos x + cos 2 x + cos 3 x)}^{2} + {(sin x + sin 2 x + sin 3 x)}^{2}}$ , then which of the following is / are correct

f (\frac{π}{12}) = \frac{\sqrt{1} + \sqrt{2} + \sqrt{3}}{\sqrt{2}}

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f (\frac{π}{5}) = \frac{3 + \sqrt{5}}{2}

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f (x) i s a n e v e n f u n c t i o n

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None of these

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Solution

The correct option is B $f (x) i s a n e v e n f u n c t i o n$
Given

$f (x) = \sqrt{{(cos x + cos 2 x + cos 3 x)}^{2} + {(sin x + sin 2 x + sin 3 x)}^{2}}$

Replace $x$ with $(- x)$

$⟹ f (- x) = \sqrt{{(cos (- x) + cos (- 2 x) + cos (- 3 x))}^{2} + {(sin (- x) + sin (- 2 x) + sin (- 3 x))}^{2}}$

$⟹ \sqrt{{(cos x + cos 2 x + cos 3 x)}^{2} + {(- (sin x + sin 2 x + sin 3 x))}^{2}}$

$⟹ \sqrt{{(cos x + cos 2 x + cos 3 x)}^{2} + {(sin x + sin 2 x + sin 3 x)}^{2}}$

$⟹ f (- x) = f (x)$

$∴ f (x)$ is an even function .

Suggest Corrections

Similar questions

y = \sqrt{(sin x + sin 2 x + sin 3 x)^{2} + (cos x + cos 2 x + cos 3 x)^{2}}

f (x) = 1 + cos x + {cos}^{2} x + \dots \infty

f (x) + f (\frac{π}{2} + x) + f (\frac{π}{2} - x) + f (π - x)

y = \sqrt{(c o s x + c o s 2 x + c o s 3 x)^{2} + (s i n x + s i n 2 x + s i n 3 x)^{2}} .

{sin}^{3} \frac{x}{2} - {cos}^{3} \frac{x}{2} = \frac{cos x \times (2 + sinx)}{3}

f (x) = \frac{1 - sin x}{sin 2 x}, x \neq \frac{π}{2}

x = \frac{π}{2},

f (\frac{π}{2})

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