Number of roots of the equation 3-cos x2- 4-2sin 8x-x- -0-5- -are

(3+cos x ) ^ 2 =4 2sin ^ 8 x where x ∈ [0,5π ] (3+cos x ) ^ 2 ∈ [4,16] as 1≤ sin x ≤ 1 , 1≤ cos x ≤ 1 4 2sin ^ 8 x ...

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Standard XII

Mathematics

Derivative of Standard Functions

Number of roo...

Question

Number of roots of the equation $(3 + cos x)^{2} = 4 - 2 {sin}^{8} x, x ϵ [0, 5 π]$ are

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Solution

${(3 + cos x)}^{2} = 4 - 2 {sin}^{8} x w h e r e x \in [0, 5 π]$

${(3 + cos x)}^{2} \in [4, 16] a s - 1 \leq sin x \leq 1, - 1 \leq cos x \leq 1$

$4 - 2 {sin}^{8} x \in [2, 4]$

$S o b o t h c a n b e e q u a l o n l y w h e n b o t h a r e e q u a l t o 4$

$T h e n sin x = 0 a n d cos x = 1 a t t h e s a m e t i m e .$

$x = (0, 2 π, 4 π}$

$W e h a v e t h r e e s o l u t i o n s$

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Number of roots of the equation (3+cosx)2=4−2sin8x,xϵ[0,5π]are

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Number of roots of the equation $(3 + cos x)^{2} = 4 - 2 {sin}^{8} x, x ϵ [0, 5 π]$ are