1. The problem statement, all variables and given/known data Hello! I am doing exercises on sinusoid functions from the beginning of Trigonometry. I hoped I understood the topic, but it seems not quite, because I don't get the results authors show as examples for one of possible answers, as there can be a few answers to the same exercise. I will be grateful for your help and explanation on what I am doing wrong. Here is the task: Show that the function is a sinusoid by rewriting it in the forms C(x) = A cos(ωx + φ) + B and S(x) = A sin(ωx + φ) + B for ω > 0. 2. Relevant equations I will show one example, and, maybe, later add more, if it is necessary. f(x) = 3√3 sin(3x) - 3cos(3x) Here is the answer from the book, and below I post how I tried to solve the task: answer from the book f(x) = 3√3 sin(3x) - 3cos(3x) = 6 sin( 3x + (11π/6) ) = 6cos( 3x + (4π/3) ) 3. The attempt at a solution My solution: f(x) = 3√3 sin(3x) - 3cos(3x) (1) rewrite the expression in the sinusoid form: f(x) = A sin(3x) cos(φ) - A cos(3x) sin(φ) Clearly, B = 0, ω = 3. (2) Find coefficients: 3√3 = A cos(φ) 3 = A sin(φ) Given Pythagorean identity we have: cos2 + sin2 = 1 Multiply both sides by A2: A2 cos2 + A2 sin2 = A2 3√32 + 32 = 36 Choose A = 6 (choosing positive 6) (3) Find φ (works both ways - take cos or sin): 3√3 = A cos(φ) 3√3 = 6 cos(φ) φ = π/6 (4) Solution: f(x) = 3√3 sin(3x) - 3cos(3x) in sin form S(x) = A sin(ωx + φ) + B f(x) = 6 sin(3x + π/6); the book gives this: f(x) = 6 sin( 3x + (11π/6) ) which is not the same: if we add π/6, both cos and sin have positive values at this angle, while at +11π/6 sin is negative. If, however, it were -11π/6, then we would have ended at the same place as if we added π/6 with both cos and sin positive. I don't see my mistakes in computations, and why my answer is wrong; and I don't see how authors achieved +11π/6 as the value of φ. To find the same formula for cos I used cofunction identities: sin(θ) = cos (π/2 - θ) In my case θ = 3x + π/6, so cos ( π/2 - 3x - π/6) = cos ( - (3x - π/3) ) = cos (3x - π/3) which is not the same as cos( 3x + (4π/3) ) for same reasons stated above in discussion on sin. -π/3 has a positive cos value (quadrant IV), while +4π/3 has a negative cos value (quadrant III). Thank you very much!