Use the half-angle formula to find sin(30).
sin(30) = sin(θ/2) θ = 60 sin(θ/2) = ± √((1-cos(θ)/2) sin(60/2) = ± √((1-cos(60)/2) sin(30) = ± √((1-cos(60)/2) sin(30) = ± √((1-1/2)/2) sin(30) = ± √(1/4) sin(30) = ± 1/2
Where is the relative maximum of the function, f(x) = 2x^3 - 11x^2 - 8x + 27?
First, take the derivative of the function to get f'(x) = 6x^2 - 22x - 8. Next, factor out the function to get f'(x) = (6x + 2)(x - 4). Next, set f'(x) = 0 and solve each part of the factor as "= 0" to get x = -1/3 and x = 4. Next, plug in values of x above, between, and below these values to see if f'(x) is positive or negative. If f'(x) is negative, the function is decreasing. If f'(x) is positive, the function is increasing. We find that the solution is increasing before x = -1/3 and decreasing after x = -1/3, meaning that the point x = -1/3 must be a maximum. The relative maximum of the function is at x = -1/3.
What values of y will make the equation, y = |7 - 4x + 5x^2|, invalid?
All values of negative y will make the equation invalid as an absolute value function creates a positive value of the solution within the brackets, meaning a negative solution is impossible.