Question

Fine the general solution
Sin(x+20) = cos3x

Answer 1

Hi ,

This is related to Trigonometric Ratios of Complementary Angles.

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As we know that two angles are said to be complementary if their

sum equals 90° .

i ) sin ( 90 -  A ) = cos A

ii ) cos ( 90 - A ) = sin A

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According to the problem ,

a ) sin ( x - 20 ) = cos ( 3x - 10 )

   ⇒  sin ( x - 20 ) = sin [ 90 - ( 3x - 10 ) ]

   ⇒   x - 20 = [ 90 - ( 3x - 10 ) ]

   ⇒   x - 20 = 90 - 3x + 10

   ⇒   x + 3x = 90 + 10 + 20

   ⇒         4x = 120

   ⇒           x = 120 / 4

    ∴          x = 30°

Or 

  sin ( x - 20 ) = cos ( 3x - 10 )

⇒ cos [ 90 - ( x - 20 ) ] = cos ( 3x - 10 ) 

⇒          90 - ( x - 20 )   = 3x - 10

⇒          90 - x + 20       = 3x  - 10

⇒                110 - x       = 3x - 10

⇒                 110 + 10   = 3x + x

⇒                          120  = 4x

                           ∴ 4x = 120

                                x = 120 / 4

                                x = 30°

I hope this helps you.

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