4.2 Trigonometric functions

From Förberedande kurs i matematik 1

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Trigonometry of right-angled triangles

In the right-angled triangle below, the ratio between the length \displaystyle a of the side opposite the angle and the length \displaystyle b of the adjacent side is called the tangent of the angle \displaystyle u, and is written as \displaystyle \tan u.

The value of the ratio \displaystyle \frac{a}{b} is not dependent on the size of the triangle, but only on the angle \displaystyle u. For different values of the angle, you can get the value of the tangent either from a trigonometric table or by using a calculator (the relevent button is usually named tan).

 x = 5\,\mbox{m} \cdot \tan 40^\circ \approx 5\,\mbox{m} \cdot 0\textrm{.}84
   = 4\textrm{.}2\,\mbox{m}\,\mbox{.}

There are two other ratios in right-angled triangles that have special names. The first is \displaystyle \cos u = b/c ("cosine of \displaystyle u") and the second is \displaystyle \sin u = a/c (" sine of \displaystyle u").

Like the tangent the ratios that define the cosine and sine do not depend on the size of the triangle, but only on the angle \displaystyle u.

Some standard angles

For some angles, namely 30°, 45° and 60°, it is relatively easy to calculate the exact values of the trigonometric functions.

 x^2 = 1^2 + 1^2
 \quad \Leftrightarrow \quad
 x = \sqrt{1^2 + 1^2} = \sqrt{2}\,\mbox{.}

Trigonometric functions for general angles

For angles less than 0° or greater than 90° the trigonometric functions are defined using the unit circle (that is the circle that has its centre at the origin and has radius 1).

The tangent function is then defined as

and the value of the tangent can be interpreted as the slope for the radial line.

 \frac{2\pi}{3} = \frac{4\pi}{6}
                = \frac{3\pi+ \pi}{6}
                = \frac{\pi}{2} + \frac{\pi}{6}.

 \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}\,\mbox{.}

Graphs of the trigonometric functions

In the last section we used a unit circle to define the cosine and sine of arbitrary angles, and we will often use the unit circle in the future, for example, to derive trigonometric relationships and solve trigonometric equations. However, there are certain characteristics of the trigonometric functions that are better illustrated by drawing their graphs.

In these graphs, we might observe several things more clearly than in the unit circle. Some examples are:

• The curves for cosine and sine repeat themselves after a change in angle of \displaystyle 2\pi, that is \displaystyle \cos (x+2\pi) = \cos x and \displaystyle \sin (x+2\pi) = \sin x. To see why this is true, note that on the unit circle \displaystyle 2\pi corresponds to a complete revolution, and after a complete revolution we return to the same point on the circle.

• The curve for the tangent repeats itself after a change in angle of \displaystyle \pi, that is \displaystyle \tan (x+\pi) = \tan x. Two angles which differ by \displaystyle \pi share the same line through the origin of the unit circle and thus their radial lines have the same slope.

• Except for a phase shift of \displaystyle \pi/2 the curves for cosine and sine are identical, that is \displaystyle \cos x = \sin (x+ \pi/2); more about this in the next section.

The curves can also be important when examining trigonometric equations. With a simple sketch, you can often get an idea of how many solutions an equation has, and where the solutions lie.

Exercises