Katie's question at Yahoo Answers: trigonometric equation

[Fernando Revilla]

Here is Katie's question:

Solve for x in the equation 1+sinx/cosx + cosx/1+sinx = 4 for 0 < x < 2pi?

Here is a link to the original question:

Solve for x in the equation 1+sinx/cosx + cosx/1+sinx = 4 for 0 < x < 2pi? - Yahoo! Answers

I'll post a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Follow the steps:

$$\dfrac{1+\sin x}{\cos x}+\dfrac{\cos x}{1+\sin x}=4\\ \dfrac{1+\sin^2x+2\sin x+\cos^2x}{\cos x(1+\sin x)}=4\\ 1+\sin^2x+2\sin x+\cos^2x=4\cos x+4\sin x\cos x$$

Using $$\cos^2x=1-\sin^2x$$ and simplifying

$$2(1+\sin x)=4(1+\sin x)\sqrt{1-\sin^2x}$$

But $$1+\sin x\neq 0$$ (because appears in a denominator of the initial equation), so

$$\sqrt{1-\sin^2x}=\frac{1}{2}$$.

Taking squares we get $$\sin x=\pm\sqrt{3}/2$$. As $$0<x<2\pi$$, we get (Edited: the following is wrong, look at the next post) $$x=\dfrac{\pi}{3},\;x=\dfrac{4\pi}{3}$$

 

[MarkFL]

I get a different result:

We are given:

$\displaystyle \frac{1+\sin(x)}{\cos(x)}+\frac{\cos(x)}{1+\sin(x)}=4$ where $\displaystyle 0<x<2\pi$

Multiply through by $\displaystyle (1+\sin(x))\cos(x)$:

$\displaystyle (1+\sin(x))^2+\cos^2(x)=4(1+\sin(x))\cos(x)$

$\displaystyle 1+2\sin(x)+\sin^2(x)+\cos^2(x)=4(1+\sin(x))\cos(x)$

Using the Pythagorean identity $\displaystyle \sin^2(x)+\cos^2(x)=1$, we have:

$\displaystyle 2+2\sin(x)=4(1+\sin(x))\cos(x)$

$\displaystyle 2(1+\sin(x))=4(1+\sin(x))\cos(x)$

Since we have $\displaystyle 1+\sin(x)\ne0$, this reduces to:

$\displaystyle 2=4\cos(x)$

$\displaystyle \cos(x)=\frac{1}{2}$ and so:

$\displaystyle x=\frac{\pi}{3},\,\frac{5\pi}{3}$

 

[Fernando Revilla]

$\displaystyle \cos(x)=\frac{1}{2}$ and so: $\displaystyle x=\frac{\pi}{3},\,\frac{5\pi}{3}$

You are right. My silly mistake: $2\pi -\frac{\pi}{3}=\frac{4\pi}{3}$ (?). Why?. I need a mathematical psychiatrist. Besides, is better to write $\cos x$ as you did instead of $\sqrt{1-\sin^2x}$, so we avoid to analyze the double sign of the root.

 

[MarkFL]

Hey, I feel somewhat guilty as Katie awarded me 10 points for posting the link to your reply...If I could transfer them to your account I would! (Smile)

 

[Fernando Revilla]

Hey, I feel somewhat guilty as Katie awarded me 10 points for posting the link to your reply...If I could transfer them to your account I would! (Smile)

No wonder. According to my solution surely Katie awarded me -10 points. :)