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So I have a problem with what I am being asked to do. Normally, when solving for the degree measures of trigonometric functions, I am presented with a fraction of rational numbers or a fraction with a rationalized radical.

What I have now is a decimal such as $\sin \theta $ = .49268329 in quadrant II. I would understand how to do this if I had theta being $\sqrt{7}/3$, because I could find the $y$ and the $r$ quite easily.

The bottom line is: Does anyone know how to solve for a trigonometric function, such as $\tan$, when an angle measure is a decimal that one must use a calculator to solve?

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What's wrong with taking $\theta = \sin^{-1}(0.4926)$ in your example? (With the adjustment that @AndréNicolas suggests) – Scaramouche Jan 13 '12 at 23:29
I get 29.5116... is that what I was supposed to get? – nmagerko Jan 13 '12 at 23:35
The sine function is symmetrical about $90^\circ$. That is, $sin(90^\circ-\theta)=\sin(90^\circ+\theta)$. So if you can figure out the angle between $0^\circ$ and $90^\circ$, you can find the second quadrant angle. A simpler equation to use is $\sin\theta=\sin(180^\circ -\theta)$. So if you know the $\theta$ in the first quadrant, you can find the second quadrant angle with the same sine. – André Nicolas Jan 13 '12 at 23:37
@AndréNicolas what is the second angle you are talking about? – nmagerko Jan 13 '12 at 23:44
@nmagerko: I meant what you referred to as the quadrant II angle. – André Nicolas Jan 13 '12 at 23:59
up vote 0 down vote accepted

Let us take your example with $\sin\theta=0.49268329$. Forget about the quadrant II stuff for a while, and think in terms of quadrant I.

Use the $\sin\theta=\frac{y}{r}$ that you referred to in the earlier post. We can always take $r=1$. If we choose $r=1$, that means that we must have $y=0.49268329$. Now if you want to figure out $x$, you can use the Pythagorean Theorem to conclude that $x^2+y^2=1^2$. So $x^2=1-y^2\approx 0.7572632$. Since $x$ is positive for the first quadrant, that gives $x\approx 0.8702087$.

Another way to think about it is that when you were told that $\sin\theta=0.49268329$, think of it as $$\sin\theta=\frac{0.49268329}{1}.$$ Then proceed exactly as you would do with $\sin\theta=\frac{\sqrt{7}}{3}$. The numbers will be messier, but fundamentally nothing has changed.

For quadrant II, the $y$ is the same, but now $x$ is the negative square root, so $x\approx -0.8702087$.

To summarize, if you are interested in the "$x$" and/or "$y$," there is no problem if you are given $\sin\theta$ or $\cos\theta$ as a decimal. You just take $r=1$.

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My calculator, when set to degrees, accepts input and provides output as decimal degrees. To convert radians to degrees, you multiply by $\frac {180}{\pi}$. Is that what you need? So if the angle is $\frac{\sqrt 7}3$ radians, that becomes about $53.050^{\circ}$. Or given $\sin \theta=.49268329$, the first quadrant angle is $\theta=0.515169$ radians or $29.517^{\circ}$. The second quadrant angle comes from $\sin 180^{\circ}-\theta=\sin \theta$, so the second quadrant angle is $150.483^{\circ}$

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What is the second quadrant angle? I don't get what that refers to. – nmagerko Jan 13 '12 at 23:40
@nmagerko: You said quadrant II, which I thought referred to the $-x +y$ quadrant of the plane, with angles $90^{\circ}$ to $180^{\circ}$. – Ross Millikan Jan 14 '12 at 1:39

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