Question

If 3sinx-5cosx = 4, Find the value of 5sinx-3cosx​

Answer 1

Answer:

Step-by-step explanation:

Given 3sinx+5cosx=5 ----(1)

On Squaring both sides of the equation (1) , we get

(3sinx+5cosx)²=5²

=> (3sinx)²+(5cosx)²+2(3sinx)(5cosx)=25

By\: algebraic\: identity:\:\\\boxed{(a+b)^{2}=a^{2}+b^{2}+2ab}

=> 9sin²x+25cos²x+2(3sinx)(5cosx)=25

=> 9(1-cos²x)+25(1-sin²x)+2(3sinx)(5cosx)=25

By\: Trigonometric\: identities:\:\\\boxed{i)sin^{2}A=1-cos^{2}A\\ii)cos^{2}A=1-sin^{2}A}

=> 9-9cos²x+25-25sin²x+2(3sinx)(5cosx)=25

=> -9cos²x-25sin²x+2(3sinx)(5cosx)=25-9-25

=> -9cos²x-25sin²x+2(3sinx)(5cosx)=-9

On multiplying both sides by (-1) , we get

=> 9cos²x+25sin²x-2(3sinx)(5cosx)=9

=> (3cosx)²+(5sinx)²-2(3cosx)(5sinx)=3²

=> (3cosx-5sinx)²=3²

By\: algebraic\: identity:\:\\\boxed{(a-b)^{2}=a^{2}+b^{2}-2ab}

=> 3cosx-5sinx =±3