Goniometric functions

In this article, I will be discussing goniometric functions, sometimes also called trigonometric functions. The word goniometry comes from Greek and means measuring angles, and trigon translates as triangle. There are four basic goniometric functions - sine, cosine, tangent and cotangent.

Basic concepts of the triangle

Goniometric functions work with angles in a triangle, so in this section we will review the concepts related to the triangle.

The figure shows a triangle that is formed by the vertices A, B, and C; it is thus the triangle ABC. Here we find three sides: AB, BC, AC. Also note that these sides are additionally named with lower case letters. This naming has a rule - opposite the vertex A we have the side a. Opposite the vertex B is the page b, and opposite the vertex C is the page c. Thus, the opposite side is always named after the vertex; the side not formed by the vertex.

Every triangle has three interior angles, which we usually denote by the Greek letters alpha α, beta β, and gamma γ. The sum of all three interior angles must always give 180 degrees. We usually have an alpha angle at the vertex A, a beta angle at B and a gamma angle at C.

Although we can use goniometric functions in some way for any triangle, we often work only with right triangles. A right triangle is a triangle that has one right angle, i.e., 90 degrees. For example, it looks like this:

A right triangle has specially named sides. The longest side is opposite the right angle and is called the hypotenuse (blue side in the picture). The two shorter sides are called the branches (red sides). This is the triangle we will be most interested in in this article.

Marking the branches in a triangle

Let's move on to the first goniometric function, the sine function. All goniometric functions show us the relationship between some angle in a triangle and the ratio of the lengths of the two sides. Usually, it is not the right angle, but the other two. The input to the goniometric function is thus the magnitude of the angle. The output is the ratio of some two sides. The different functions differ depending on which sides they work with.

The sine function works with the reciprocal of the sag and the hypotenuse. What is an adjacent and opposite branch relative to a given angle is shown in the following figure.

In the figure we are working with the angle beta, the angle at the vertex B. The black side is the hypotenuse, nothing changes on it. The side highlighted in red c is the adjacent overhang because it is adjacent to the angle beta. The blue highlighted side b is the opposite side of the overhang because it is opposite the beta angle. Importantly, these terms always refer to an angle. If we look at the figure from the perspective of the gamma angle, we get this result:

The sine function

The sine of the alpha angle is equal to the ratio of the length of the opposite branch to the length of the hypotenuse. What does this mean? If we calculate (on a calculator, for example) the sine of the alpha angle, we get the value of the ratio

$$\sin(\alpha)=\frac{\mbox{ The length of the opposite branch }}{\mbox{ The length of the hypotenuse }}$$

Let's try this on this triangle and the angle beta: It's a nice triangle, the lengths of the sides are: |a| = 5, |b| = 3 and |c| = 4. The opposite branch to the angle beta is the side b, the hypotenuse is the side a. Calculate the quotient b/a, so 3/5, which is 0,6. The sine of the angle beta is then equal to 0.6. The angle beta is $36^\circ 52^\prime$, if you plug this angle into the calculator and calculate the sine, you get just 0.6 (after a little rounding). You can check the result in the calculator for calculating the value of the sine function.

What it's good for

This is good if you know one angle and the length of one side and need to calculate the remaining sides.

The first thing to do is to write down what you actually know. We know the angle alpha. We know that the sine of this angle is equal to the ratio of the opposite side (which is the side a) to the hypotenuse (the side b). Written mathematically:

$$\sin(\alpha)=\frac{|a|}{|b|}$$

We know, or can calculate, the sine of the angle; we know the length of the side a. The only thing we don't know is the length of side b. So we will try to isolate this variable. First we multiply the equation by the side length b. We get

$$|b|\cdot\sin(\alpha)=|a|$$

and now we just divide the equation by sin(α) to get the desired shape with the side length b on the left side:

$$|b|=\frac{|a|}{\sin(\alpha)}.$$

We know the side length of a, it is equal to three. The sine of thirty degrees is equal to one half. Add:

$$|b|=\frac{3}{0{,}5}=6$$

The side b has a length of six.

The cosine function

Cosine is very similar to the sine function. The terminology around the cosine function is the same as in the previous section, so let's move on to the definition:

The cosine of the angle alpha is equal to the ratio of the length of the adjacent branch to the length of the hypotenuse. So if we calculate the cosine of the alpha angle on the calculator, we get the ratio of the

$$\cos(\alpha)=\frac{\mbox{ the length of the adjacent branch }}{\mbox{ the length of the hypotenuse }}.$$

The difference between the sine and cosine is clearly shown in the following figure:

Both functions work with the hypotenuse of the triangle, which remains black. The sine then works with the opposite offsets, which - relative to the angle alpha - is the red side BC. The cosine works with the adjacent offsets, which is the blue side AB.

Let us return to the example described in the last chapter. We had to calculate the length of the hypotenuse if we knew the length of the side a. Let's now try to calculate the length of the side c, if we know the angle alpha and the length of the hypotenuse we just calculated in the previous example: b = 6.

What do we know? The alpha angle is 30 degrees, the side length b is equal to six, and the cosine gives us the ratio of the adjacent overhang to the hypotenuse. The adjacent eave is the side c, which is the side whose length we want to find. We write:

$$\cos(\alpha)=\frac{|c|}{|b|}$$

We need to isolate |c| from the equation. We do this by multiplying the equation by the length of the side b, which leaves us with just the length of the side c on the right side:

$$|b|\cdot\cos(\alpha)=|c|$$

We just flip the sides:

$$|c|=|b|\cdot\cos(\alpha)$$

On the right hand side we have expressions that we either know or can calculate. The cosine of thirty degrees is approximately 0.866(see the calculator for calculating the cosine function). Add to the equation:

$$|c|=6\cdot0{,}866=5{,}196$$

This is the length of the side c. As we will see below, the cosine of thirty degrees is a table value that is equal to

$$\cos(30^\circ)=\frac{\sqrt{3}}{2}(=0{,}866025\ldots),$$

so we can refine the previous length to:

$$|c|=6\cdot\frac{\sqrt{3}}{2}=3\sqrt{3}.$$

Tangent and cotangent (cotangent)

There are two other goniometric functions, tangent and cotangent. The main difference from the previous goniometric functions is that tangent and cotangent only work with branches, they do not work with the hypotenuse.

The tangent of the alpha angle is equal to the ratio of the length of the opposite eave to the length of the adjacent eave. We usually denote the tangent by either tg or tan.

$$\tan(\alpha)=\frac{\mbox{ The length of the opposite branch }}{\mbox{ The length of the adjacent eaves }}$$

The cotangent of the alpha angle is equal to the ratio of the length of the adjacent eave to the length of the opposite eave. We usually denote the cotangent by cot or cotan.

$$\cot(\alpha)=\frac{\mbox{ The length of the adjacent eaves }}{\mbox{ The length of the opposite branch }}$$

Again, the terminology is the same as in the previous sections and other work with functions is the same. Let's go back to the previous example and try to calculate the side length c just from knowing the magnitude of the angle alpha and the side length a, which is a = 3.

The tangent is the ratio of the opposite branch to the adjacent branch, so mathematically written it will look like this:

$$\tan(\alpha)=\frac{|a|}{|c|}$$

We need to calculate the side length c, so we need to isolate the expression |c| from this equation. We multiply by |c| and divide by $\tan(\alpha)$, similar to the previous examples.

$$\begin{eqnarray} \tan(\alpha)&=&\frac{|a|}{|c|}\quad/\cdot |c|\\ |c|\cdot\tan(\alpha)&=&|a|\quad/:\tan(\alpha)\\ |c|&=&\frac{|a|}{\tan(\alpha)} \end{eqnarray}$$

Add to the right side, where the tangent of thirty degrees is approximately equal to 0.5773(see the calculator for calculating the value of the tangent function).

$$|c|=\frac{3}{0{,}5773}=5{,}196.$$

There is a table value for the tangent of thirty degrees, it is true that:

$$\tan(30^\circ)=\frac{\sqrt{3}}{3},$$

so we can calculate the previous calculation exactly:

$$|c|=\frac{3}{\frac{\sqrt{3}}{3}}=3\cdot\frac{3}{\sqrt{3}}=\frac{9}{\sqrt{3}}.$$

Tabular values

Sine, cosine, tangent and cotangent have nice resultant values for some nice angles. Here is a basic overview of them:

$$ \LARGE \begin{matrix} &\sin&\cos&\tan&\cot\\ 0^\circ&0&1&0&\times\\ 30^\circ&\frac12&\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{3}&\sqrt{3}\\ 45^\circ&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&1&1\\ 60^\circ&\frac{\sqrt{3}}{2}&\frac12&\sqrt{3}&\frac{\sqrt{3}}{3}\\ 90^\circ&1&0&\times&0 \end{matrix} $$

Links

Related concepts here on the web:

• Oriented angle
• Radian

Further information on other sites:

• Goniometry and trigonometry - matfyz thesis
• Textbook by Martin Krynicki
• Introduction to Goniometry/Trigonometry on maths.cz