SOLUTION: Express &#8722;5.9sin(x)+6.6cos(x) in the form ksin(x+&#981;) K= &#981;= Note: &#981; should be given in radians in the interval &#8722;&#960;<&#981;<&#960;

(1)     -5.9sin(x)+6.6cos(x) in the form ksin(x+ϕ)

That makes us think of the right side of the identity:

cos(x+ϕ) = cos(x)cos(ϕ) - sin(x)sin(ϕ)

multiplied by some constant k:

(2)     k*cos(x+ϕ) = k[cos(x)cos(ϕ) - sin(x)sin(ϕ)]

Let's get 

(1)     -5.9sin(x)+6.6cos(x)

in the form of the right side of (2)

        k[cos(x)cos(ϕ) - sin(x)sin(ϕ)]  

(1)     -5.9sin(x)+6.6cos(x) =
        
        cos(x)(6.6) - sin(x)(5.9) 

Let A be some positive non-zero constant. Then by multiplying
then dividing by A

        

We find A and ϕ such that

 and 

We see that ϕ must be in QIV because that's where the
cosine is positive and the sine is positive.  We draw
an angle in QI to represent ϕ.  

 

We can find A by the Pythagorean theorem:






We can find ϕ from 



using the inverse tangent feature on a calculator
in radian mode:

ϕ = 0.7294565922

So 
        

now becomes

       

which is equivalent to

      

or

      

Edwin