Why are the trig functions called sine, cosine and tangent?

By Martin McBride, 2023-04-14
Tags: triangle sine cosine tangent secant arcsine
Categories: gcse trigonometry


You probably know of the three main trigonometry functions - sine, cosine and tangent. You might also know of some extra trig functions called secant, cosecant and cotangent. And you might have seen that the inverse trig functions are sometimes referred to as arcsine, arccosine, etc.

But where do these names come from?

In this article, we will look at the origins of these names.

If you are not familiar with chords, tangents, and secants of a circle, take a look at the parts of a circle article.

Primary trig functions - sine, tangent, secant

When we think about the main trig functions, we usually think of sine, cosine and tangent. Those are the functions that are most often used to solve trigonometry problems.

But historically the sine, tangent and secant functions were considered the primary functions.

Why is this? If we form a triangle inside a unit circle, with an angle a at the centre, then the sine, tangent and secant functions will tell us the lengths of each of the three sides in terms of a. Each function is named after the side it relates to.

Sine function

The sine function is related to the chord of a circle. A chord is a line between two points on the circumference of the circle. Here is an example of a chord of a circle:

Chord

The word sine is an old term for the chord of a circle. It originates from the Sanskrit word for the string of a bow (as in a bow and arrow) because the chord and arc of a circle looks quite like a bow:

Chord

To understand how the sine function relates to a chord of a circle, we can draw a triangle within a unit circle:

Sine function

Here, the right-angled triangle POQ has an angle a at the centre. The hypotenuse, OP, has length 1 because it is a radius of the unit circle. The side opposite to angle a has length x.

The definition of the sine function is:

Sine function

Substituting the values for opposite and hypotenuse gives:

Sine function

Now if we draw a second, congruent triangle ROQ, we can see that the line PR forms a chord of the circle:

Sine function

So sin a tells us the length x, which is the length of the side of the triangle that makes up part of the chord PR. We call it the sine function because sine means chord.

In fact, the length x is equal to half the length of the chord. The sine function was sometimes called the half-chord function, although that term is rarely used these days.

Tangent function

As you might expect, the tangent function relates to the tangent of a circle. A tangent to a circle is a line that touches the circumference of a circle, without crossing it. Here is an example:

Tangent

Once again we can draw a triangle in the unit circle to discover how the tangent function relates to the tangent of a circle:

Tangent

This triangle TOS is not quite the same as the one we drew for the sine function. The previous triangle had a hypotenuse of length 1, this triangle has the side adjacent to angle a of length 1.

The opposite side ST has length y.

The diagram also shows a tangent to the circle, the line SU. ST is part of that tangent line.

The definition of the tangent function is:

Tangent function

Substituting the values for opposite and adjacent gives:

Tangent function

So tan a tells us the length y. This is the length of the side of the triangle ST that makes up part of the tangent SU.

We call it the tangent function because of this.

Secant function

The secant function relates to the secant of a circle, again as you would expect. A secant is a line that crosses the circumference of a circle in two places. A secant is similar to a chord, except that it extends outside the circumference:

Secant

This time we draw the same triangle TOS that we drew for the tangent example:

Secant

In this diagram the hypotenuse SO has length z. The adjacent side OT has length 1 because it is a radius of the unit circle.

The line SV is a secant to the circle, so the hypotenuse SO is part of the secant.

The secant function is the reciprocal of the cosine function, so it is defined as:

Secant function

Substituting the values for hypotenuse and adjacent gives:

Secant function

So sec a tells us the length SO, which is the length of the side of the triangle that makes up part of the secant SV. So we call it the secant function.

Secondary trig functions - cosine, cotangent, cosecant

The names of the secondary trig functions are formed by adding the prefix co to the name of one of the primary functions. This indicates that the function is based on the complementary angle.

In a right-angled triangle, the two acute angles a and b are called complementary angles:

Complementary angle

Angles a and b always add up to 90 degrees.

In the case of trig functions, the primary functions are based on the angle at the centre of the circle (that we have been calling a). The secondary functions are based on the complementary angle b.

Cosine function

This is the diagram we used before to illustrate the sine function, but this time the complementary angles b is shown too:

Cosine function

We have previously seen the equation for the sine function:

Sine function

The cosine function applies to the same sides PQ and OP, but this time it relates them to angle b:

Cosine function

To be clear, the value of sin a tells us the length PQ (ie x)in terms of the angle a. The value of cos b also tells us the value of x, but in terms of the angle b.

Of course, side PQ is opposite angle a, but it is adjacent to angle b, so substituting the side names gives us the usual formula for cosine:

Cosine function

Cotangent and cosecant functions

The cotangent function can be found in a similar way. We won't go through it in detail, but the formula for angle cot b uses the same sides as tan a but switches opposite and adjacent:

Cotangent function

Similar for the cosecant:

Cosecant function

Inverse trig functions

The inverse trig functions allow us to find the angle from two sides, for example, if:

Sine function

Then:

Arcsine function

This inverse sine function is sometimes called arcsine of arcsin. Similarly, the inverse tangent can be called arctangent or arctan and so on.

Why is this? Well, if we measure an angle in radians, at the centre of a unit circle, then the length of the arc it creates is equal to the angle. Here is an illustration for arcsin:

Cosine function

The inverse sine of x is equal to the length of the arc PW. It is also equal to the angle a too, of course.

See also



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