Given that A(sin θ + cos θ) + B(cos θ − sin θ) ≡ 4 sin θ, find the values of the constants A and B.
Since this must be true for all values of θ, and cos and sin are distinct functions, no non-zero multiple of cosθ could ever be equal to 4 sinθ for all values of θ. Therefore, the overall multiple of cosθ on the left-hand-side must be 0.
Therefore, Acosθ + Bcosθ ≡ 0and (A+B) cosθ ≡ 0so A = - B
We can then plug this back into the equation to solve for A:A(sinθ + cosθ) - A(cosθ - sinθ) ≡ 4sinθAsinθ - (-Asinθ) = 4sinθ [the cosθ terms cancel one another out]Asinθ + Asinθ = 4sinθ2A = 4A = 2B = - A = - 2
DB
Related Maths A Level answers
All answers ▸a) Solve the following equation by completing the square: x^(2)+ 6x + 1= 0. b) Solve the following equation by factorisation: x^(2) - 4x - 5 = 0 c) Solve the following quadratic inequality: x^(2) - 4x - 5 < 0 (hint use your answer to part b)
Find the coordinates of the stationary points for the curve y = x^4 - 2*x^2 + 5.
a) Integrate ln(x) + 1/x - x to find the equation for Curve A b) find the x coordinate on Curve A when y = 0.
Find the derivative and following function and hence find the value of coordinates for when the function is at a stationary point: