Question

if u can, do it as soon as possible

Answer 1

Here is your solution :

Trigonometric identities used in this question.

=> cos∅ = √( 1 - sin²∅ )

Algebric identities used in this question,

=> ( a - b)² = a² + b² - 2ab

Given,

=> 2cos∅ + sin∅ = 1

=> 2[ √( 1 - sin²∅ ) ] + sin∅ = 1

=> 2[ √( 1 - sin²∅ ) ] = ( 1 - sin∅ )

=> √( 1 - sin²∅ ) = ( 1 - sin∅ ) / 2

=> ( 1 - sin²∅ ) = [ ( 1 - sin∅ ) / 2 ]²

=> ( 1 - sin²∅ ) = [ ( 1 + sin²∅ - 2 sin∅ ) / 4 ]

=> 4( 1 - sin²∅ ) = ( 1 + sin²∅ - 2 sin∅ )

=> 4 - 4 sin²∅ = 1 + sin²∅ - 2 sin∅

=> - 4 sin²∅ - sin²∅ + 4 - 1 + 2 sin∅ = 0

=> - 5 sin²∅ + 3 + 2 sin∅ = 0

=> 5 sin²∅ - 2 sin∅ - 3 = 0

=> 5 sin²∅ - 5 sin∅ + 3 sin∅ - 3 = 0

=> 5 sin∅ ( sin∅ - 1 ) + 3 ( sin∅ - 1 ) = 0

=> ( sin∅ - 1 ) ( 5 sin∅ + 3 ) = 0

=> ( sin∅ - 1 ) = 0 ÷ ( 5 sin∅ + 3 )

=> sin∅ - 1 = 0

•°• sin∅ = 1

" Or "

=> ( sin∅ - 1 ) ( 5 sin∅ + 3 ) = 0

=> ( 5 sin∅ + 3 ) = 0 ÷ ( sin∅ - 1 )

=> ( 5 sin∅ + 3 ) = 0

=> 5 sin∅ = - 3

•°• sin∅ = ( -3 / 5 )

Hence, sin∅ = 1 or ( -3/5 )

Now,

=> cos∅ = √( 1 - sin²∅ )

If, sin∅ = 1

=> cos∅ = √( 1 - 1² )

=> cos∅ = √( 1 - 1 )

=> cos∅ = √0

•°• cos∅ = 0

Now,

= 4 cos∅ + 3 sin∅

= ( 4 × 0 ) + 3 × 1

= 0 + 3

= 3

When, sin∅ = ( -3 / 5 )

=> cos∅ = √( 1 - sin²∅ )

=> cos∅ = √[ 1 - ( -3 / 5 )² ]

=> cos∅ = √[ 1 - ( 9/25 ) ]

=> cos∅ = √[ ( 25 - 9 ) / 25 ]

=> cos∅ = √( 16 / 25 )

•°• cos∅ = 4/5

Now,

If , cos∅ = 4/5

= 4 cos∅ + 3 sin∅

= 4 ( 4/5 ) + 3 ( -3/5 )

= ( 16/5 ) - ( 9/5 )

= ( 16 - 9 ) /5

= 7/5

Hence, the required answers are 3 and ( 7/5 ).

Hope it helps !!