Express 5.6 sin ( x ) + 1.2 cos ( x ) 5.6 sin ( x ) + 1.2 cos ( x ) in the form k sin ( x + ϕ ) k sin ( x + ϕ )

Question

Express  5.6sin(x)+1.2cos(x)  in the form  ksin(x+ϕ)

K=_____

ϕ= _____

Answer 1

We begin by using the addition formula for the sin() function.

k·sin(x + φ) = k·sin(x)cos(φ) + k·sin(φ)cos(x)

k·sin(x + φ) = k·cos(φ)sin(x) + k·sin(φ)cos(x)

From this, we see that k·cos(φ) = 5.6 and k·sin(φ) = 1.2 must be true. Simple division shows that

(k·sin(φ))/k·cos(φ) = tan(φ)

(k·sin(φ))/k·cos(φ) = 1.2/5.6

Then, tan(φ) = 1.2/5.6 so φ = tan^-1(1.2/5.6) ≈ 12.09°. Knowing φ, we can determine k using k·sin(φ) = 1.2 such that:

k·sin(φ) = 1.2

k = 1.2/sin(φ)

k = 1.2/sin(tan^-1(1.2/5.6))

k ≈ 5.73

Thus, k ≈ 5.73 and φ ≈ 12.09°.