General solution to ( 3 − 1 ) cos ⁡ x + ( 3 +...

Maliyah Robles

Maliyah Robles

Answered

2022-07-11

General solution to ( 3 1 ) cos x + ( 3 + 1 ) sin x = 2

Answer & Explanation

Charlee Gentry

Charlee Gentry

Expert

2022-07-12Added 19 answers

hint is: a cos θ + b sin θ = c
Given: ( 3 1 ) cos θ + ( 3 + 1 ) sin θ = 2
Let ( 3 1 ) = r cos α and ( 3 + 1 ) = r sin α
Then r cos α cos θ + r sin α sin θ = 2 r cos ( θ α ) = 2 cos ( θ α ) = 2 r
Now, r = ( 3 1 ) 2 + ( 3 + 1 ) 2 = 8 = 2 2
Thus, cos ( θ α ) = 1 2 = cos π 4
Also, tan α = 3 + 1 3 1 = tan ( π 2 π 3 + π 4 ) α = 5 π 12
Thus: ( θ α ) = 2 n π ± π 4 . Giving, θ = 2 n π ± π 4 + 5 π 12 .

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