The solution of the equation kcos⁡x−3sin⁡x=k+1 is possible if

Here, k cos x - 3 sin x = k +1 could be rewritten askk2+9cos⁡x−3k2+9sin⁡x=k+1k2+9⇒ cos⁡(x+ϕ)=k+1k2+9where cos⁡ϕ=kk2+9and sin⁡ϕ=3k2+9which posses solution only, if−1≤k+1k2+9≤1⇒k+1k2+9≤1⇒(k+1)2≤k2+9⇒k2+2k+1≤k2+9⇒k≤4Hence, the interval in which kcos x - 3sin x = k +1 admits solution for k is (−∞,4]