Question

1+4sinx+3cosx=?IITian needed

Answer 1

Hey there!!!
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 1 + 4 sinx + 3 cosx

multiplying and dividing by 5 :

= 5 (1/5 + 4/5 sinx  + 3/5 cos x)

Let y = sin^-1(4/5)
=> y = cos^-1(3/5)

= 5(1/5 + siny . sinx + cosy . cosx)               
= 5(1/5 + cos (x -y) )           ......[ using identity, cos(x-y)= cosx . cosy + sinx . siny]
= 1 + 5 cos(x-y)
= 1 + 5 cos ( x - cos^-1(3/5) )

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Hope it helps!!!

Answer 2

1 + 4sinx +3cosx = ?

Actually
-√(a²+ b²) ≤ asinx + bcosx ≤ √(a² + b²)
It means , range of value of (asinx+bcosx) € [-√(a²+b²), √(a²+b²)]

So, the value of
4sinx + 3cosx belongs to [ -√(4²+3²),√(4²+3²)] e.g [-5, 5]
Hence, value of 1+4sinx+3cosx €[-4,6]

-4 ≤ 1+4sinx + 3cosx ≤ 6

Hence, all real number belongs to inerval [-4,6] are the values of (1+4sinx+3cosx)