Question
Solve the equation:sin(x) + cos(x) = 2·sin(x + α).
Then the value of α is:Solution
Given the equation: sin x + cos x = 2 · sin(x + α) We start by rewriting the left-hand side using a standard identity. Multiply and divide the expression sin x + cos x by √2: This gives us: √2 · (1/√2 · sin x + 1/√2 · cos x) Now recall the identity: cos(π/4) = sin(π/4) = 1/√2 So we rewrite the expression as: √2 · (cos(π/4) · sin x + sin(π/4) · cos x) This is in the form: √2 · sin(x + π/4) Therefore, the left-hand side becomes: √2 · sin(x + π/4) = 2 · sin(x + α) Now divide both sides by 2: (√2 / 2) · sin(x + π/4) = sin(x + α) This implies that sin(x + α) = sin(x + π/4) multiplied by √2 / 2. But for the two sine expressions to be equal for all x , the coefficients must match exactly. So we equate the inner angles directly: x + π/4 = x + α → α = π/4
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