If-sin-2cos-1-Prove-that-2sin-cos-2-

Question Number 7683 by new phone last updated on 08/Sep/16

If \sin θ + 2 \cos θ = 1

Prove that 2 \sin θ - \cos θ = 2

Commented by sandy_suhendra last updated on 08/Sep/16

s i n θ +^{} 2 c o s θ = 1

{(s i n θ + 2 c o s θ)}^{2} = 1^{2}

{s i n}^{2} θ + 4 s i n θ c o s θ + 4 {c o s}^{2} θ = 1

4 s i n θ c o s θ = 1 - {s i n}^{2} θ - 4 {c o s}^{2} θ

{(2 s i n θ - c o s θ)}^{2}

= 4 {s i n}^{2} θ - 4 s i n θ c o s θ + {c o s}^{2} θ

= 4 {s i n}^{2} θ - (1 - {s i n}^{2} θ - 4 {c o s}^{2} θ) + {c o s}^{2} θ

= 5 {s i n}^{2} θ + 5 {c o s}^{2} θ - 1

= 5 ({s i n}^{2} θ + {c o s}^{2} θ) - 1

= 5 - 1 = 4

s o (2 s i n θ - c o s θ) = \pm 2