Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 $r=1$.

share|cite|improve this answer

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 $150.483^{\circ}$

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.