# Thread: Given equation Find angles

1. ## Given equation Find angles

Which angles $\displaystyle \alpha$ will satisfy the equation $\displaystyle \5\sin^2\alpha\ + \25sin\alpha\ - 2.4375 = 0$ ?

I tried the following to no avail.
$\displaystyle \5sin\alpha\(sin\alpha + 1)\ - 2.4375 = 0$
Don't know where to go from here. Maybe use the quadratic formula? Been a while since I've done these.

Thanks

2. ## Re: Given equation Find angles

?? I see you have considered the Quadratic Formula? Do it! $\displaystyle \sin(\alpha) =$ ??

3. ## Re: Given equation Find angles

Thanks TK, I did the quadratic formula:
$\displaystyle \frac{\-25\pm\sqrt{625 + 20(2.4375)}}{10}$

Took the acrsin of 0.09... Still not getting it.

4. ## Re: Given equation Find angles

What are you not getting? That looks good. You threw out -5.095669... That was right on track.

arcsin is good, but not quite all the answer. There is a twin over there around pi!

Note sin(x) = x is a very good approximation for very small x. Don't be too freaked out when you hit the arcsin button and for a moment you think nothing happened.

5. ## Re: Given equation Find angles

Apologies, I didn't understand when you said There is a twin over there around pie. So the answer would be 1.7pi? I'm thinking $\displaystyle \sin\alpha=0.09..$Sorry for being an idiot.

6. ## Re: Given equation Find angles

In general if you have the goniometric equation:
$\displaystyle \sin(x)=a$
Two solutions:
$\displaystyle \Leftrightarrow x=\arcsin(a)+2k\pi$
$\displaystyle \Leftrightarrow x=\pi - \arcsin(a)+2k\pi$

7. ## Re: Given equation Find angles

Originally Posted by freestar
Which angles $\displaystyle \alpha$ will satisfy the equation $\displaystyle \5\sin^2\alpha\ + \25sin\alpha\ - 2.4375 = 0$ ?

I tried the following to no avail.
$\displaystyle \5sin\alpha\(sin\alpha + 1)\ - 2.4375 = 0$
Don't know where to go from here. Maybe use the quadratic formula? Been a while since I've done these.

Thanks
Let $\displaystyle \displaystyle x = \sin{\alpha}$ so that your equation becomes $\displaystyle \displaystyle 5x^2 + 25x - 2.4375 = 0$. Solve for $\displaystyle \displaystyle x$ and use this to solve for $\displaystyle \displaystyle \alpha$.

8. ## Re: Given equation Find angles

I replaced $\displaystyle \sin\alpha$ with $\displaystyle x$
$\displaystyle 5x^2 + 25x - 2.4375 = 0$
$\displaystyle x^2 + x - 0.4875 = 0$
$\displaystyle \frac{-1\pm\sqrt{1+4(0.4875)}}{2}$
solving for $\displaystyle \sin\alpha$
$\displaystyle \sin^-^1(0.35877) = 21.0247$degrees

9. ## Re: Given equation Find angles

Originally Posted by freestar
I replaced $\displaystyle \sin\alpha$ with $\displaystyle x$
$\displaystyle 5x^2 + 25x - 2.4375 = 0$
$\displaystyle x^2 + x - 0.4875 = 0$
$\displaystyle \frac{-1\pm\sqrt{1+4(0.4875)}}{2}$
solving for $\displaystyle \sin\alpha$
$\displaystyle \sin^-^1(0.35877) = 21.0247$degrees
When you divide everything by 5 you should get $\displaystyle \displaystyle x^2 + 5x - 0.4875 = 0$.

10. ## Re: Given equation Find angles

Sorry. Silly mistake!
Correction:

$\displaystyle x^2 + 5x - 0.4875 = 0$
$\displaystyle \frac{-5\pm\sqrt{25+4(0.4875)}}{2}$
solving for $\displaystyle \sin\alpha$
$\displaystyle \sin^-^1(0.09567) = 5.4899$degrees
What would be next?

11. ## Re: Given equation Find angles

Originally Posted by freestar
Sorry. Silly mistake!
Correction:

$\displaystyle x^2 + 5x - 0.4875 = 0$
$\displaystyle \frac{-5\pm\sqrt{25+4(0.4875)}}{2}$
solving for $\displaystyle \sin\alpha$
$\displaystyle \sin^-^1(0.09567) = 5.4899$degrees
What would be next?
You are probably expected to give the answer in radians, but anyway. You have the answer that corresponds to the first quadrant. Where else is the sine function positive? Where does the sine function repeat itself?

12. ## Re: Given equation Find angles

Yes sir, I want to find the answer in pi and that is what I am having difficulty figuring out because when I try to convert it, I just get some decimal number times pi.
The other angle would be 180-5.4899 = 174.5 degrees. Sine is positive in the top right and left (1st and 2nd quadrant).
How can I write them as proper pi values? Really appreciate the help eh.

13. ## Re: Given equation Find angles

Originally Posted by freestar
Yes sir, I want to find the answer in pi and that is what I am having difficulty figuring out because when I try to convert it, I just get some decimal number times pi.
The other angle would be 180-5.4899 = 174.5 degrees. Sine is positive in the top right and left (1st and 2nd quadrant).
How can I write them as proper pi values? Really appreciate the help eh.
You should be well aware that $\displaystyle \displaystyle 2\pi^C = 360^{\circ} \implies \frac{2\pi}{360}^C = 1^{\circ} \implies 1^{\circ} = \frac{\pi}{180}^C$.

So to convert degrees to radians, you need to multiply by $\displaystyle \displaystyle \frac{\pi}{180}$. Since you have decimal approximations for your degrees, I expect you don't need to leave your radian answers in terms of $\displaystyle \displaystyle \pi$.

14. ## Re: Given equation Find angles

I know how to convert them to radians. I guess I was just trying to figure out if I could get it to be a nice fraction time pi or something. I'll just leave the radians in decimals. Thanks a lot! Problem solved.