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Question
Solve the following equations:
sin θ + cos θ = `sqrt(2)`
Solution
Divide each term by `sqrt(2)`
`1/sqrt(2) sin theta + 1/sqrt(2) cos theta = sqrt(2)/sqrt(2)`
`sin pi/4 sin theta + cos pi/4 cos theta` = 1
`cos theta * cos pi/4 + sin theta * sin pi/4` = 1
`cos (theta - pi/4)` = 1
`cos (theta - pi/4)` = cos θ
The general solution is
`theta - pi/4` = 2nπ, n ∈ Z
θ = `2"n"pi + pi/4`, n ∈ Z
θ = `(8"n"pi + pi)/4`, n ∈ Z
θ = `(8"n" + 1) pi/4`, n ∈ Z
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