find maximum value of 4(sinx+acosx)(sinx+bcosx) - (a+b)^2
If a and b are constants, then the max occurs where the derivative is zero.
That is where tan(2x) = (a+b)/(ab-1)
I guess you can find some values of a and b where that is easy, or express sinx and cosx for such an angle.
To find the maximum value of the expression 4(sinx+acosx)(sinx+bcosx) - (a+b)^2, we need to differentiate it with respect to x and set the derivative equal to zero. Let's start by expanding the expression:
4(sinx+acosx)(sinx+bcosx) - (a+b)^2
= 4(sin^2x + asinxcosx + bsinxcosx + abcos^2x) - (a+b)^2
Next, we differentiate this expression with respect to x:
d/dx [4(sin^2x + asinxcosx + bsinxcosx + abcos^2x) - (a+b)^2]
= 4(2sinxcosx + a(cos^2x-sin^2x) + b(cos^2x-sin^2x) - 2abcosxcosx)
Now, we set this derivative equal to zero and solve for x:
4(2sinxcosx + a(cos^2x-sin^2x) + b(cos^2x-sin^2x) - 2abcosxcosx) = 0
2sinxcosx + a(cos^2x-sin^2x) + b(cos^2x-sin^2x) - 2abcosxcosx = 0
Simplifying this equation further, we get:
2sinxcosx + a(1-sin^2x) + b(1-sin^2x) - 2abcosxcosx = 0
2sinxcosx + a - asin^2x + b - bsin^2x - 2abcosxcosx = 0
Now, we can gather like terms:
(a + b - a - b) + (2sinxcosx - asin^2x - bsin^2x - 2abcosxcosx) = 0
(2sinxcosx - asin^2x - bsin^2x - 2abcosxcosx) = 0
Since sin^2x + cos^2x = 1, we can replace sin^2x in the equation:
(2sinxcosx - a(1-cos^2x) - bsin^2x - 2abcosxcosx) = 0
(2sinxcosx - a + acos^2x - bsin^2x - 2abcosxcosx) = 0
Now, let's simplify further:
2sinxcosx - a + acos^2x - bsin^2x - 2abcosxcosx = 0
(2sinxcosx - 2abcosxcosx) + (acos^2x - a - bsin^2x) = 0
2cosx(sinxcosx - abcosx) + a(cos^2x - 1) - bsin^2x = 0
Now, we can factor out common terms:
2cosx(sinxcosx - abcosx) + a(cos^2x - 1 - sin^2x) = 0
2cosx(sinxcosx - abcosx) + a(-sin^2x - cos^2x) = 0
2cosx(sinxcosx - abcosx) - a = 0
Finally, we can solve for x:
2cosx(sinxcosx - abcosx) - a = 0
2cosx = a / (sinxcosx - abcosx)
cosx = a / (2sinxcosx - abcosx)
To find the maximum value, we substitute this value of cosx back into the original expression and evaluate it.
To find the maximum value of the expression 4(sinx+acosx)(sinx+bcosx) - (a+b)^2, we can use calculus.
Step 1: Expand the expression:
4(sinx+acosx)(sinx+bcosx) - (a+b)^2
= 4(sin^2(x)+abcos^2(x) + asinx + bcosx) - (a+b)^2
Step 2: Simplify the expression:
= 4sin^2(x) + 4abcos^2(x) + 4asinx + 4bcosx - (a+b)^2
= 4sin^2(x) + 4ab(1-sin^2(x)) + 4asinx + 4bcosx - a^2 - 2ab - b^2
Step 3: Rearrange the terms:
= (4a - 4ab)sin^2(x) + (4b)cos^2(x) + (4a)sinx + (4b)cosx - a^2 - 2ab - b^2
Step 4: Let's find the derivative of this expression with respect to x:
d/dx [(4a - 4ab)sin^2(x) + (4b)cos^2(x) + (4a)sinx + (4b)cosx - a^2 - 2ab - b^2]
To simplify, let's consider:
f(x) = (4a - 4ab)sin^2(x) + (4b)cos^2(x) + (4a)sinx + (4b)cosx - a^2 - 2ab - b^2
Now, find the derivative of f(x) with respect to x:
f'(x) = [(4a - 4ab) * 2sin(x) * cos(x)] + [(-4b) * 2cos(x) * sin(x)] + [4a * cos(x)] + [(-4b) * sin(x)]
= 4(2a - 2ab)sin(x)cos(x) - 4b(sin^2(x) - cos^2(x)) + 4acos(x) - 4bsin(x)
= 8(a - ab)sin(x)cos(x) - 4b(sin^2(x) - cos^2(x)) + 4acos(x) - 4bsin(x)
Step 5: Set the derivative equal to zero to find the critical points:
8(a - ab)sin(x)cos(x) - 4b(sin^2(x) - cos^2(x)) + 4acos(x) - 4bsin(x) = 0
Step 6: Solve for x using trigonometric identities:
Analyze the equation using trigonometric identities like sin^2(x) + cos^2(x) = 1 and sin(2x) = 2sin(x)cos(x).
Simplify the equation:
8(a - ab)sin(x)cos(x) - 4b(sin^2(x) - cos^2(x)) + 4acos(x) - 4bsin(x) = 0
8(a - ab)sin(x)cos(x) - 4b(sin^2(x) - (1 - sin^2(x))) + 4acos(x) - 4bsin(x) = 0
8(a - ab)sin(x)cos(x) - 4b(2sin^2(x) - 1) + 4acos(x) - 4bsin(x) = 0
8(a - ab)sin(x)cos(x) - 8b(sin^2(x) - 1/2) + 4acos(x) - 4bsin(x) = 0
8(a - ab)sin(x)cos(x) - 8b(1/2 - sin^2(x)) + 4acos(x) - 4bsin(x) = 0
8(a - ab)sin(x)cos(x) - 4b + 8bsin^2(x) + 4acos(x) - 4bsin(x) = 0
8(a - ab)sin(x)cos(x) + 8bsin^2(x) + 4acos(x) - (4bsin(x) + 4b) = 0
8sin(x)[(a - ab)cos(x) + bsin(x)] + 4acos(x) - 4b = 0
From this equation, we can solve for x by analyzing the different factors and trigonometric identities.