Range of a function involving trigonometric functions

[tellmesomething]

Homework Statement

Range of $\displaystyle \left| \dfrac{(√(cosx)-√(sinx))(√(cosx)+√(sinx))} {3(cosx+sinx)} \right|$

Relevant Equations

None

I resolved the numerator to $cosx-sinx$
We get $$mod\frac{cosx-sinx} {3(cosx+sinx)} $$
If we divide the numerator and denominator by cosx we get
$$mod\frac{1-tanx} {3(1+tanx)}$$(eq1)
We know that tan(π/4-x) is same as $\frac{1-tanx} {1+tanx}$
So re writing eq1 we get
$$mod\frac{tan(π/4-x} {3}$$
As we know tangent function can take any value from -∞ to+∞
Considering the modulus function we can conclude that the range is 0 to ∞

However thats not the case ofcourse. I graphed it on desmos and while the original question lies on the graph of this simplified tan function, its range is bounded

Please tell me where I went wrong?

 

[Aurelius120]

Aren't the graph and range correct though?

 

[tellmesomething]

Aren't the graph and range correct though?

They aren't. The original expression's graph is only the red part.

The expression I simplified and got (in terms of tan) is the whole blue part.

 

[tellmesomething]

@fresh_42 Please taje a look at this if you have some time.

 

[Ibix]

What's $\sqrt{\cos(\pi)}$?

 

[tellmesomething]

What's $\sqrt{\cos(\pi)}$?

Oh. So ill have to restrict the domain π/4 - x should not be equal to places where sin and cos get negative

 

[Aurelius120]

They aren't. The original expression's graph is only the red part.

The expression I simplified and got (in terms of tan) is the whole blue part.

I see.
It could be because the square root function restricts the value of $\cos x$ and $\sin x$

 

[tellmesomething]

I see.
It could be because the square root function restricts the value of $\cos x$ and $\sin x$

I am sorry I dont know how I overlooked that. Thankyou for your Help.

 

[tellmesomething]

What's $\sqrt{\cos(\pi)}$?

But doesnt that just restrict the domain from 0 to π/2 ?

 

[Ibix]

But doesnt that just restrict the domain from 0 to π/2 ?

No, just to anywhere both sin and cos are positive.

 

[fresh_42]

@fresh_42 Please taje a look at this if you have some time.

I see a pattern. You do the right algebra on the formulas but forget what has already been hidden in the original expressions. The last time you solved $ae^{2x} + be^x+c=0$ and investigated the discriminant. However, you forgot that $e^x>0$ regardless of the solution of the quadratic. Now, you did the correct algebra again, but by using $(\sqrt{a}+\sqrt{b})\cdot (\sqrt{a}-\sqrt{b})=a-b$ you forgot that you lost $a,b\geqq 0.$ I'm not sure which kind of practice is appropriate in such a situation. My guess is, that your mathematical enthusiasm leads you directly into algebra, e.g. being happy that you correctly identified the formula that has to be used. I understand this, I have the same tendency to jump into the problem and calculate. That often leads to situations where I do not see the obvious. Hence, the only advice I can give both of us is to wait a moment and inspect the problem before doing any algebra. The only problem: who reminds us of this advice?

 

[tellmesomething]

I see a pattern. You do the right algebra on the formulas but forget what has already been hidden in the original expressions. The last time you solved $ae^{2x} + be^x+c=0$ and investigated the discriminant. However, you forgot that $e^x>0$ regardless of the solution of the quadratic. Now, you did the correct algebra again, but by using $(\sqrt{a}+\sqrt{b})\cdot (\sqrt{a}-\sqrt{b})=a-b$ you forgot that you lost $a,b\geqq 0.$ I'm not sure which kind of practice is appropriate in such a situation. My guess is, that your mathematical enthusiasm leads you directly into algebra, e.g. being happy that you correctly identified the formula that has to be used. I understand this, I have the same tendency to jump into the problem and calculate. That often leads to situations where I do not see the obvious. Hence, the only advice I can give both of us is to wait a moment and inspect the problem before doing any algebra. The only problem: who reminds us of this advice?

Have to remind our own self