MHB Answerer's question at Yahoo Answers regarding a trigonometric equation

[MarkFL]

Here is the question:

PRE-CALC QUESTION!? HELP !?

0<x<2π
3sin^2x - 2sinx - 3 = 0

You can use a calculator and there's only 2 answers

Here is a link to the question:

PRE-CALC QUESTION!? HELP !? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[MarkFL]

Re: answerer's question at Yahoo! Answers regarding trigonometric equation

Hello answerer,

We are given to solve:

$$3\sin^2(x)-2\sin(x)-3=0$$ where $$0<x<2\pi$$

Recognizing that we have a quadratic in $\sin(x)$, we may use the quadratic formula to state:

$$\sin(x)=\frac{-(-2)\pm\sqrt{(-2)^2-4(3)(-3)}}{2(3)}=\frac{2\pm\sqrt{40}}{6}=\frac{1\pm\sqrt{10}}{3}$$

Since we require $$-1\le\sin(x)\le1$$ we discard the positive root, and we are left with:

$$\sin(x)=\frac{1-\sqrt{10}}{3}$$

Hence:

$$x=\sin^{-1}\left(\frac{1-\sqrt{10}}{3} \right)$$

Since this is less than zero, we need to add $$2\pi$$ to get the equivalent angle in the required interval:

$$x=2\pi+\sin^{-1}\left(\frac{1-\sqrt{10}}{3} \right)$$

Now, this is the 4th quadrant solution, but we should observe there is also a 3rd quadrant solution, given by:

$$x=\pi-\sin^{-1}\left(\frac{1-\sqrt{10}}{3} \right)$$

Note: this comes from the identity $$\sin(\pi-\theta)=\sin(\theta)$$.

If we are to use a calculator to obtain decimal approximations, then:

$$x=2\pi+\sin^{-1}\left(\frac{1-\sqrt{10}}{3} \right)\approx5.47828834852818$$

$$x=\pi-\sin^{-1}\left(\frac{1-\sqrt{10}}{3} \right)\approx3.9464896122412$$

To answerer and any other guests viewing this topic, I invite and encourage you to post other trigonometry questions here in our http://www.mathhelpboards.com/f12/ forum.

Best Regards,

Mark.