trigonometric function


The trigonometric functions are most easily understood in the context of a circle in the Cartesian plane with its center at the origin, and in which angles are always measured from the \(x\)-axis. (Positive angles are measured in an anti-clockwise direction, and negative angles are measured in a clockwise direction.)

Figure 1: Angle in the Cartesian plane.

Any such angle is associated with a right triangle whose hypotenuse is the radius, with the side adjacent to the angle on the \(x\)-axis and the side opposite the angle parallel to the \(y\)-axis. In this context the trig functions sine, cosine, and tangent are defined as ratios of the sides of the triangle:

\[\sin \varphi = \frac{\mbox{opp}}{\mbox{rad}},\,\,\,\cos \varphi = \frac{\mbox{adj}}{\mbox{rad}},\,\,\,\tan \varphi = \frac{\mbox{opp}}{\mbox{adj}}\]

Three additional trig functions, cosecant, secant, and cotangent, are defined as the multiplicative inverses of the first three:

\[\csc \varphi = \frac{\mbox{rad}}{\mbox{opp}},\,\,\,\sec \varphi = \frac{\mbox{rad}}{\mbox{adj}},\,\,\,\cot \varphi = \frac{\mbox{adj}}{\mbox{opp}}\]

So we have,

\[\csc \varphi = \frac{1}{\sin \varphi},\,\,\,\sec \varphi = \frac{1}{\cos \varphi},\,\,\,\cot \varphi = \frac{1}{\tan \varphi}\]

Notice also that \(\tan \varphi = \displaystyle\frac{\sin \varphi}{\cos \varphi}\).

Values of the Trig Functions

One of the first things to notice about the trig functions is that they are periodic: they take on the same values cyclically because for any angle \(\varphi\) the triangle that defines the trig functions will be the same for \(\varphi\) plus any multiple of \(2\pi\) radians (360°). Notationally, \(\sin \varphi = \sin (\varphi + 2k\pi)\) for any integer \(k\), and likewise for the other five trig functions.

Another useful landmark is that for certain angles the values of the functions are particularly easy to calculate. At 0 (or any multiple of \(2\pi\) radians) the triangle ‘collapses’—the adjacent side is equal to the radius and the opposite side becomes 0. It is easy to see then that

\[\sin 0 = 0,\,\,\,\cos 0 = 1,\,\,\,\tan 0 = 0\]

The same thing happens at \(\pi\) radians (180°), except in this case the adjacent side is negative, so while \(\sin \pi\) and \(\tan \pi\) are still both 0, we have \(\cos \pi = -1\)

The following are all of the other ‘easy’ values for sine, cosine, and tangent that one should be aware of. (The keen student should verify each one for herself.)

\[\sin \frac{\pi}{2} = 1,\,\,\,\cos \frac{\pi}{2} = 0,\,\,\,\tan \frac{\pi}{2} = undef\]

\[\sin \frac{3\pi}{2} = -1,\,\,\,\cos \frac{3\pi}{2} = 0,\,\,\,\tan \frac{3\pi}{2} = undef\]

\[\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2},\,\,\,\cos \frac{\pi}{3} = \frac{1}{2},\,\,\,\tan \frac{\pi}{3} = \sqrt{3}\]

\[\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2},\,\,\,\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2},\,\,\,\tan \frac{\pi}{4} = 1\]

\[\sin \frac{\pi}{6} = \frac{1}{2},\,\,\,\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2},\,\,\,\tan \frac{\pi}{6} = \frac{\sqrt{3}}{3}\]

Because the denominator goes to 0 for tangent at both \(\displaystyle\frac{\pi}{2}\) and \(\displaystyle\frac{3\pi}{2}\) we say that both \(\tan \displaystyle\frac{\pi}{2}\) and \(\tan \displaystyle\frac{3\pi}{2}\) are undefined.

Graphs of the Trig Functions

The graph of the sine function is familiar, being a smooth ‘wave’-shaped curve that achieves maximum and minimum values of 1 and -1, and passes through the \(x\)-axis at multiples of \(\pi\). Here we picture a complete cycle from \(x=0\) to \(x=2\pi\).

Figure 2: Graph of the sine function.

The full graph of sine is just the same curve repeated endlessly in both directions.

The graph of cosine—as you might expect from the symmetry of the situation—is exactly the same except that it is displaced or ‘out of phase’ by a distance of \(\displaystyle\frac{\pi}{2}\) (i.e., by 90°).

Figure 3: Graph of the cosine function.

Here we see it passing through the points \((0,1)\), \((\frac{\pi}{2},0)\), \((\pi,-1)\), \((\frac{3\pi}{2},0)\), and \((2\pi,1)\).

The graph of the tangent function is quite different because of the infinite discontinuities at odd multiples of \(\displaystyle\frac{\pi}{2}\).

Figure 4: Graph of the tangent function.

As with other kinds of functions, the graphs of the trig functions can be affected by dilations and translations. In the case of the sine and cosine functions these are especially important, as the dilations affect the period and the amplitude of the wave.

Students sometimes find the trigonometric functions alien and somewhat disconnected from what they think of as ‘regular’ math, but in fact their importance to mathematics, especially applied mathematics, is central and surprisingly far reaching. Through the technique of Fourier analysis any continuous function can be arbitrarily closely approximated by a sum of trigonometric functions, a fact which lies at the heart of the digital revolution in technology. Mastering the trig functions is therefore a major stepping stone in one's study of mathematics.