# Minimum value of $\cos^2\theta-6\sin\theta \cos\theta+3\sin^2\theta+2$

Recently I was solving one question, in which I was solving for the smallest value of this expression

$$f(\theta)=\cos^2\theta-6\sin\theta \cos\theta+3\sin^2\theta+2$$

My first attempt:

\begin{align} f(\theta) &=3+2\sin^2\theta-6\sin\theta \cos\theta \\ &=3(1-2\sin\theta \cos\theta)+2\sin^2\theta \\ &=3(\sin\theta-\cos\theta)^2 + 2\sin^2\theta \end{align}

Hence the minimum value of $f(\theta)=2\sin^2\theta$ when $\theta=\pi/4$ hence minimum value of $f(\theta)=1$.

But then again I tried to do question differently by making substitutions in order to change the whole $f(\theta)$ in the form of $\cos x+\sin x$ Then $f(\theta)$ came out to be $$f(\theta)= 4-(\cos(2\theta)+3\sin(2\theta))$$ The minimum value of this expression is surely $$(f(\theta))_{min}=4-\sqrt{10}$$

Can anybody explain me algebraically why my first method gave the wrong result?

• "Hence the minimum value of $f(\theta)=2\sin^2(\theta)$ when $\theta=\pi/4$ hence minimum value of $f(\theta)=1$" It has no sense ! – Surb Jun 29 '16 at 18:22
• @Surb Why? If $\sin(\theta)-\cos(\theta)$ carry any value then its square would get added up on to the total terms as this is contained in a square and hence the minimum value would come when the $\sin(\theta)-\cos(\theta)$=0. Is this wrong? – Harsh Sharma Jun 29 '16 at 18:24
• and so ? this will not implies that $f(\theta)$ will be minimal at $\theta=\pi/4$ (as you can see). – Surb Jun 29 '16 at 18:28
• Disregard the prior comment – Mark Viola Jun 29 '16 at 19:00

$f(\theta) = 4 - \sqrt{10}$ is correct.

so what is the error here:

$f(\theta) = 3(\sin \theta-\cos\theta)^2 + 2\sin^2\theta$

That part is a true statement but then you say.

Hence the minimum value of $f(θ)=2\sin^2θ,$ when $θ=π/4$ hence minimum value of $f(θ)=1.$

It is a logical jump that was a step too far.

$f(\theta) = 3(\sin \theta-\cos\theta)^2 + k$

it would be okay to zero out the term in parentheses. It must be greater than or equal to zero, so set it to zero.

$f(\theta) = 3(\sin \theta-\cos\theta)^2 + 2\sin^2\theta$

Minimizing either term doesn't minimize the sum.

Furthermore, when you say the minimum of $f(θ)=2\sin^2θ,$ occurs when $θ=\pi/4$ that is just wrong.

• OK because the term outside too is a function of $\theta$, hence I can't use that thing. Thanks a lot sir. – Harsh Sharma Jun 29 '16 at 19:06

hint: Alternatively, using Lagrange Multiplier we have:

$x = \cos \theta, y = \sin \theta\implies f(x,y) = x^2-6xy+3y^2+2, x^2+y^2 = 1\implies f(x,y) = (x^2+y^2)+2y^2+2 - 6xy = 3-6xy+2y^2 \implies f_x = -6y= 2\lambda x, f_y = 4y-6x= 2\lambda y\implies4\lambda y - 6\lambda x = 2\lambda^2y\implies 4\lambda y+18y=2\lambda^2y\implies y(\lambda^2-2\lambda-9) = 0\implies y = 0$ or $\lambda = 1\pm \sqrt{10}$ . Can you continue at this point?

• Sir, I am in high school and we have not been taught about Lagrange multiplier, but I still appreciate your effort for helping me. Thanks a lot. – Harsh Sharma Jun 29 '16 at 20:32
• You are welcome, and infact the LM is easy to learn. All you need is to solve the nabla equation $\nabla f = \lambda \nabla g$ – DeepSea Jun 29 '16 at 20:33
• I am pretty curious to learn that. Would really appreciate, if you could help me with any handout or other information. And again Thanks a lot. – Harsh Sharma Jun 29 '16 at 20:34

$f'(\theta)=3(sin\theta-cos\theta)^2 + 2sin^2\theta = 6 (\sin\theta -\cos\theta )(\cos\theta + \sin\theta) + 4 \sin \theta \cos\theta$

$=6 (\sin^2 \theta - \cos^2 \theta) + 4 \sin \theta \cos\theta =-6\cos(2\theta) +2\sin(2\theta)\ge 0 \implies \frac{2}{6} \tan (2\theta)\ge 1$

$\implies \tan(2\theta)\ge 3 \implies 2\theta\ge \arctan(3) \implies \theta \ge \arctan(3)/2$

What that tells us is that first derivative is bigger than $0$ when $\theta \ge \arctan(3)/2$ that is function is growing and after that its smaller than $0$ - meaning its decreasing so we know that the function has a minimum in point $\theta = \arctan(3)/2$

With that $$f(\theta)=\cos^2\theta-6\sin\theta \cos\theta+3\sin^2\theta+2=\frac{1+\cos(2\theta)}{2} - 3\sin(2\theta) +3\frac{1-\cos(2\theta)}{2} + 2=4-\cos(2\theta)-3\sin(2\theta) = 4-\frac{1}{\sqrt{10}} - 3\frac{3}{\sqrt{10}}=4 - \sqrt{10}$$

Since we know that $\cos^2 (2\theta)+\sin^2 (2\theta)=1 \implies 1+9=1/\cos^2 (2\theta) \implies cos^2(2\theta)=\frac{1}{10}$

• Your answer is truly correct mathematically and I had also taken the derivative of the function. But can you just suggest me the reason why taking up $sin\theta=cos\theta$ to make the expression $(\sin\theta-\cos\theta)^2=0$ would be wrong as this always comes as a positive terms and would get on to add to other factors increasing their value? – Harsh Sharma Jun 29 '16 at 18:29
• Because making the first addend 0 does not mean that the sum of those two will be minimal. You need to figure in the second addend also! ($2\sin^2(2\theta)$) in your case. Example of what a flawed logic this is would be $x+ (x+2)^{100}$ You made the first 0 so you say the sum is zero but here its $2^{100}$ – daniels_pa Jun 29 '16 at 18:40
• But if I choose $x=-2$ won't it still make the thing minimum i.e. x=-2 – Harsh Sharma Jun 29 '16 at 18:47
• No. $x+(1+x)^2$ is not minimum when x is $0$ or when $x=-1$ because $x^2 +3x +1=0$ when we solve quadratic formula $\frac{-3\pm \sqrt{9-4}}{2}=\frac{-3\pm \sqrt{5}}{2}$ – daniels_pa Jun 29 '16 at 18:55
• No problem do the derivative yourself on the $x+(x+2)^{100}$ and you will see that the minimum is not in -2 (but quite close to it) – daniels_pa Jun 29 '16 at 19:13