How do you find the minimum of the equation sin^2(x) + 4sin(x)?
The first step to solving this problem is to treat it as a normal quadratic equation; if you are struggling with comparing our equation to a normal quadratic, try substituting sin(x) = y into the equation as shown:
sin2(x) + 4sin(x) becomes y2 + 4y
Even though our equation does not equal 0 we can still use the 'complete the square' method to help us find the minimum, after applying this method our equation becomes:
(y + 2)2 - 4
From this we can substitue y for sin(x), giving:
(sin(x) + 2)2 - 4
To find the minimum of our equation we have to take in to account the fact that sin(x) has a range of -1 to 1, which limits (sin(x) + 2) to a range of 1 to 3.
From this you should be able to deduct that the smallest value of (sin(x) + 2)2 - 4 is -3. This occurs when sin(x) = -1.
(-1 + 2)2 - 4 = 12 - 4 = -3
Hence the minimum of sin2(x) + 4sin(x) is -3.
Related Maths A Level answers
All answers ▸Derive the quadratic formula. From it, write down the determinant and explain, how is it related to the roots of a quadratic equation.
A block of mass 5kg is on a rough slope inclined at an angle of 30 degrees to the horizontal, it is at the point of sliding down the slope. Calculate the coefficient of friction between the block and the slope.
Identify the stationary points of f(x)=3x^3+2x^2+4 (by finding the first and second derivative) and determine their nature.
Sketch the graph of x^2+y^2-6x-4y=23