Ifa cos3 - -3a cos - sin2 -m and a sin3 - -3a cos2 - sin - -n- then m-n23-m-n23 is equal to

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Range of Trigonometric Expressions

Ifa cos3 α +3...

Question

If
$a c o s^{3} α + 3 a c o s α s i n^{2} α = m a n d a s i n^{3} α + 3 a c o s^{2} α s i n α = n, t h e n (m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}} i s e q u a l t o$

$2 a^{2}$

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$2 a^{\frac{1}{3}}$

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$2 a^{\frac{2}{3}}$

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$2 a^{3}$

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Solution

The correct option is C
$2 a^{\frac{2}{3}}$

From the given relations, we get
$m + n = a c o s^{3} α + 3 a c o s α {sin}^{2} α + 3 a {cos}^{2} α sin α + a {sin}^{3} α$
$m + n = a (cos α + sin α)^{3}$
$S i m i l a r l y m - n = a (cos α - sin α)^{3} ∴ (m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}} = a^{\frac{2}{3}} [(cos α + sin α)^{2} + (cos α - sin α)^{2}] = a^{\frac{2}{3}} [1 + 2 sin α cos α + 1 - 2 sin α cos α] = 2 a^{\frac{2}{3}}$

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Similar questions

$I f a c o s^{3} α + 3 a c o s α s i n^{2} α = m a n d a s i n^{3} α + 3 a c o s^{2} α s i n α = n, t h e n (m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}} i s e q u a l t o$

Q. Q27

27 . I f a \cos^{3} θ + 3 a \sin^{2} θ \cos θ = m a n d a \sin^{3} θ + 3 a \sin θ \cos^{2} θ = n, p r o v e t h a t {(m + n)}^{2 / 3} + {(m - n)}^{2 / 3} = 2 a^{2 / 3}

$I f t a n β = \frac{n s i n α c o s α}{1 - n s i n^{2} α}, t h e n t a n (α - β) i s e q u a l t o$

Q. If

α + β = 90^{o}

and

α = 2 β

, then

{cos}^{2} α + {sin}^{2} β

equals to

Q. If

∣ ∣ ∣ \begin{matrix} {sin}^{2} α & {cos}^{2} α {cos}^{2} α & {sin}^{2} α \end{matrix} ∣ ∣ ∣ = 0, α ϵ (0, π)

, then the values of

α

are :

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If a cos3 α+3a cos α sin2 α=m and a sin3 α +3a cos2 α sin α=n, then (m+n)23+(m−n)23 is equal to

The correct option is C 2a23 From the given relations, we get m+n=a cos3α+3a cos α sin2α+3a cos2 α sin α+a sin3α m+n=a(cosα+sinα)3 Similarly m−n=a(cos α−sin α)3∴ (m+n)23+(m−n)23=a23[(cos α+sin α)2+(cos α−sin α)2]=a23[1+2sinαcosα+1−2sinαcosα]=2a23

If
$a c o s^{3} α + 3 a c o s α s i n^{2} α = m a n d a s i n^{3} α + 3 a c o s^{2} α s i n α = n, t h e n (m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}} i s e q u a l t o$