# Touch up on Trig

I have forgotten a few things about trigonometry and angles.
I have this trig equation, $\sin \theta = \frac{200\text{ dyn}}{224\text{ dyn}}$

What exactly are the steps of getting the angle, $\theta$.

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Is there supposed to be an equal sign in your equation? What is $dyn$? – Jim Belk Jun 14 '11 at 16:45
I'm assuming this says: sin(theta)=200/224 in which case you use the inverse sine (aka the arcsine function): asin(200/224)=theta. Because if you asin() both sides... asin(sin(theta))=asin(200/224) and asin(sin(anything)) = anything. – Matt Razza Jun 14 '11 at 16:45
Without an equals sign, that's not an equation you've got there. – Milosz Wielondek Jun 14 '11 at 16:45
Oh yes! sorry about that. – Daniel Pendergast Jun 14 '11 at 16:59
@Jim Belk: I assume that $\text{dyn}$ is the abbreviation for the unit of force equal to $1\text{gcm/s}^2$. See Wikipedia, dyne. – Américo Tavares Jun 14 '11 at 17:11

I do not know whether you are measuring angles in radians or in degrees. So I will assume degrees. If you know about radians, I am sure that you can make the requisite adjustments.

First note that there is not a single answer to your problem. For if $\sin \theta= a$, then we also have $\sin(180^\circ-\theta)=a$. And if $\sin\theta=a$, then $\sin(\theta+360n)=a$ for any integer $n$.

But for any number $a$ such that $0\le a \le 1$, there is exactly one $\theta$ between $0^\circ$ and $90^\circ$ such that $\sin\theta=a$.

On most older calculators, entering $a$ and then pressing the $\sin^{-1}$ button will give you the (unique) $\theta$ between $-90^\circ$ and $90^\circ$ which solves the equation $\sin\theta=a$. On many newer calculators, you press the $\sin^{-1}$ button then put in $a$. Make sure your calculator is set to degrees.

When I do this, I get roughly $79.155937$. Try it on your calculator, or in the calculator program on your computer. It makes sense that the angle is not far from $90^\circ$, since the sine of a $90$ degree angle is $1$, and $220/224$ is not far from $1$.

However, as I pointed out earlier, there are other angles whose sine is $220/224$. An important one that you are likely to bump into in geometric work is $180$ minus the number we just computed. This is roughly $100.84406$ degrees.

And, as was pointed out, there are infinitely many angles whose sine is $220/224$. However, there is only one from $0$ to $90$ degrees, and only one from $90$ degrees to $180$ degrees, and we have found them.

You might want to check, by calculating the sine of each of these angles, that each of them has the right sine, at least to the limits of calculator accuracy.

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@yunone: Thanks for the edit! – André Nicolas Jun 14 '11 at 18:49
Thank you! sin -1 works perfectly. – Daniel Pendergast Jun 14 '11 at 19:54

The solution to the equation $$\sin \theta \;=\; \frac{220}{224}$$ is $$\theta \;=\; \arcsin\left(\frac{220}{224}\right)$$ where $\arcsin$ denotes the inverse sine function.

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Thank you very much! – Daniel Pendergast Jun 14 '11 at 19:55
$\tfrac{220}{224}=\tfrac{55}{56}$ – Elements in Space Dec 24 '12 at 9:26