Trigonometry Tutorial 2
We should now investigate the rules for these functions in other quadrants as shown in Panel 7. In the first case, we see that the angle to be dealt with is \(\alpha = \theta - 180^\circ\). Now both \(r\) and \(p\) are negative. So we have the \(\sin(\theta) = (-r)/q = -\sin(\alpha) = -\sin (\alpha) = -\sin (\theta - 180^\circ)\). Similarly \(\cos(\theta) = (-p)/q = -\cos(\alpha) = -\cos (\theta - 180^\circ)\). And \(\tan (\theta) = (-r)/(-p) = r/p = \tan (\alpha) = \tan(\theta - 180^\circ)\).
Notice in this case only the tangent is positive. Using your calculator determine the sine, cosine and tangent of \(190^\circ\).
The angle we are concerned with is \(190^\circ - 180^\circ = 10^\circ\).
\(\sin(10^\circ) = 0.1736\)
\(\cos(10^\circ) = 0.9848\)
\(\tan(10^\circ) = 0.1763\).
Therefore,
\(\sin(190^\circ) = - 0.1736\)
\(\cos(190^\circ) = - 0.9848\)
\(\tan(190^\circ) = 0.1763\)
Finally, in the bottom of Panel 7 is the last case. The angle of concern \(\theta\) is now \((360^\circ - \alpha)\), and \(p\) and \(q\) are positive while \(r\) is negative. As before, we get the following relations given below. Notice only the cosine is positive.
\(\sin(\theta) = -\frac{r}{q} = \sin(\alpha) = -\sin(360^\circ - \theta)\)
\(\cos (\theta) = \frac{p}{q} = \cos (\alpha) = \cos (360^\circ - \theta)\)
\(\tan (\theta) = - \frac{r}{p} = - \tan(\alpha) = - \tan (360^\circ - \theta)\)
There is a simple rule by which you can remember all of these results, summarized in Panel 8. Notice in the first quadrant, all the functions are positive; in the second quadrant, only the sine is positive; in the third, only the tangent is positive; and in the fourth quadrant, only the cosine is positive. The little mnemonic at the right, called the CAST Rule, tells you which function is positive in each quadrant. And the c, a, s and t stand for cosine, all, sine and tangent.
Let us now look at a graph of these functions. But before we examine it in detail, let's see what we can learn just from inspection.
Look at Panel 9. From our definition, \(\sin(q) = r/q\), you can see that sine of \(0\) is \(0\) since \(r\) will be 0. The value of sine will rise as \(\theta\) increases and reach a value of \(1\) when \(q = 90^\circ\), since then \(r\) will be equal to \(q\). Obviously, then, the sine function is one which increases from \(0\) to a maximum value of \(1\) as \(\theta\) increases from \(0^\circ\) to \(90^\circ\). You can see this plotted in Panel 10a.
From \(90^\circ\) to \(180^\circ\), it decreases back to \(0\) but remains positive as we saw earlier.
From \(180^\circ\) to \(360^\circ\), you remember, it was always negative, and you can readily see, it has a minimum value of \(-1\) at \(270^\circ\). Beyond \(360^\circ\), of course, it just repeats. The function is what we call an oscillatory function, and in your studies in physics, you will find it most important to appreciate this property, particularly in the study of waves and alternating current in electricity and electronics.
How does the cosine function behave? Panel 10 again shows you. At \(0^\circ\), \(q = p\) and so the value is \(1\). At \(90^\circ\), \(p = 0\), and so \(\cos(90^\circ) = 0\). This is shown in Panel 10b. You can work out the rest of it for yourself. You see that the cosine function is exactly the same as the sine function if you slide the sine function graph back \(90^\circ\). This says that \(\sin(90^\circ + q) = \cos (\theta)\).
The tangent curve looks quite different. Panel 11 shows that \(\tan(0) = r/p = 0\), but \(\tan(90^\circ) = r/p = \infty\) since \(r\) is finite but \(p\) has gone to \(0\). Panel 11 shows this function. The tangent function is a repeating one but not oscillatory.
One last point which you will find used many times are the approximate values of these functions when \(\theta\) is very small. Look at Panel 12. This is like our other figures except that we have put the triangle into the sector of a circle of radius \(q\). You can readily see that as \(\theta\) gets very small, \(p\) and \(q\) become very nearly equal and so the cosine approaches the value \(1\) as we already know. But in this approximation, \(r/q\) and \(r/p\) are very nearly equal. These, of course, are the sine and the tangent of \(\theta\). So we may say that for very small values of \(\theta\), \(\sin(\theta)\) is very nearly equal to \(\tan(\theta)\). Indeed, if you look back below Panel 7, you will see we looked up the functions for \(\theta = 10^\circ\). Even for an angle of this size, you can see that \(\cos(10^\circ)\) is within \(1.5\%\) of \(1\); and \(\sin(10^\circ)\) and \(\tan(10^\circ)\) are equal to within about \(2\) parts in \(170\), or about \(1.2\%\).
We can even go further. You will remember the relation that the arc length \(s\) is equal to the radius \(q\) multiplied by the angle measured in radians \(\theta\); that is, \(s = q \theta\). Now in the small angle limit I have been talking about, \(s\) and \(r\) are nearly equal. Therefore, for the ratio \(r/q\), we can use \(s/q\) and we immediately see that, for small angles, the sine, the tangent, and the angle \(\theta\) itself measured in radians are all nearly equal. We need to stress that this is only true if \(\theta\) is measured in radians, where you will recall that \(2\pi\) radians was equal to \(360^\circ\).
You have now learned the simple rules of trigonometry which, with practice, should equip you for your introductory science courses. Try this Trigonometry Quiz to check your understanding?