The general solution of the trigonometric equation is given by
- A.;
- B.;
- C.;
- D.None of the above
Show answer and solution
Correct option: C
Solution
Rewrite sin x + cos x = 1 as √2 sin(x + π/4) = 1, so sin(x + π/4) = 1/√2 = sin(π/4).
The general solution of sin θ = sin α is θ = nπ + (−1)ⁿ α.
So x + π/4 = nπ + (−1)ⁿ (π/4), giving x = nπ + (−1)ⁿ (π/4) − π/4.
This covers both families: when n is even, x = nπ + π/4 − π/4 = nπ (giving x = 0, 2π, …) and when n is odd, x = nπ − π/4 − π/4 = nπ − π/2 (giving x = π/2, 5π/2, …), which are exactly the solutions. Options 1 and 2 each capture only one family, so option 3 is the complete general solution.
The general solution of sin θ = sin α is θ = nπ + (−1)ⁿ α.
So x + π/4 = nπ + (−1)ⁿ (π/4), giving x = nπ + (−1)ⁿ (π/4) − π/4.
This covers both families: when n is even, x = nπ + π/4 − π/4 = nπ (giving x = 0, 2π, …) and when n is odd, x = nπ − π/4 − π/4 = nπ − π/2 (giving x = π/2, 5π/2, …), which are exactly the solutions. Options 1 and 2 each capture only one family, so option 3 is the complete general solution.
[1981]