Trigonometry/In simple terms

This page is intended as a simplified introduction to trigonometry. (This article is not always correctly formulated in mathematical language.)

Simple introduction

Introduction to Angles

If you are unfamiliar with angles, where they come from, and why they are actually required, this section will help you develop your understanding.

An angle between two lines in a 2-dimesional plane

In two dimensional flat space, two straight lines that are not parallel lines meet at an angle. Suppose you wish to measure the angle between two lines exactly so that you can tell a remote friend about it: draw a circle with its center located at the meeting of the two lines, making sure that the circle is small enough to cross both lines, but large enough for you to measure the distance along the circle's edge, the circumference, between the two cross points. Obviously this distance depends on the size of the circle, but as long as you tell your friend both the radius of the circle used, and the length along the circumference, then your friend will be able to reconstruct the angle exactly.

Triangle Definition

We know that triangles have three sides: each side is a line: if we choose any two sides of the triangle, then we have chosen two lines, which are not parallel, and must therefore meet at an angle. There are three ways that we can pick the two sides, so a triangle must have three angles: hence tri-angles, shortened to triangle.

This argument does not always work in reverse though. If you give me three angles, for example three right angles, I cannot make a triangle from them. This is also a problem with sides: I can give you three lengths that do not make the sides of a triangle: your height, the height of the nearest tree, the distance from the top of the tree to the center of the sun.

Triangle Ratios

Angles are not affected by the length of lines: an angle is invariant under transformations of scale (if you make the triangle bigger, its angles won't change). Given the three angles of a known triangle, you can draw a new triangle that is similar to the old triangle but not necessarily congruent. Two triangles are called similar if they have the same angles as each other, they are congruent if they have the same size angles and matching sides have the same length. Given a pair of similar triangles, if the lengths of two matching sides are equal, then the lengths of the other sides are also equal, and so the triangles are congruent. More generally, the ratios of the lengths of matching pairs of sides between two similar triangles are equal. In particular, a triangle can be divided into 4 congruent triangles by connecting the mid-points on the sides of the original triangle together, each congruent triangle is similar on half scale to the original triangle.

Consequently, angles are useful for making comparisons between similar triangles.

Right Angles

An angle of particular significance is the right-angle: the angle at each corner of a square or a rectangle. Indeed, a rectangle can always be divided into two triangles by drawing a line through opposing corners of the rectangle. This pair of triangles has interesting properties. First, the triangles are identical. Second, they each have one angle which is a right-angle.

Imagine sitting at a table drawing a rectangle on a sheet of paper, and then dividing the rectangle into a pair of triangles by drawing a line from one corner to another. Perhaps the two triangles are different? A friend could sit on the other side of the table and draw the exact same thing by re-using your lines. The triangle nearest you has been produced by the exact same process as the triangle nearest your friend, therefore the two triangles are congruent, that is, identical. Try cutting a rectangle into two triangles to check.

Each triangle of this pair of triangles has one right angle: the rectangle has four right-angles, we split two of them by drawing from corner to corner, the remaining two were distributed equally to the identical triangles, so each triangle got one right-angle.

Thus every right angled triangle is half a rectangle.

A rectangle has four sides of two different lengths: two long sides and two short sides. When we split the rectangle into two identical right angled triangles, each triangle has a long side and a short side from the rectangle as well as a copy of the split line, which has a Greek name: "hypotenuse". In an aside, trigonometry has been defined as 'Many cheerful facts about the square of the hypotenuse'.

So the area of a right angled triangle is half the area of the rectangle from which it was split. Looking at a right angled triangle we can tell what the long and short sides of that rectangle were, they are the sides, the lines, that meet at a right angle. The area of the rectangle is the long side times the short side. The area of a right angled triangle is therefore half as much.

Right Triangles and Measurement

How to bisect an angle

It is possible to bisect any angle using only circles (which can be drawn with a compass) and straight lines by the following procedure:

1. Call the vertex of the angle O. Draw a circle centered at O.
2. Mark where the circle intersects each ray. Call these points A and B.
3. Draw circles centered at A and B with equal radii, but make sure that these radii are large enough to make the circles intersect at two points. One sure way to do this is to draw line segment AB and make the radius of the circles equal to the length of that line segment. On the diagram, circles A and B are shown as near-half portions of a circle.
4. Mark where these circles intersect, and connect these two points with a line. This line bisects the original angle.

A proof that the line bisects the angle is found in Proposition 9 of Book 1 of the Elements.

Given a right angle, we can use this process to split that right angle indefinitely to form any binary fraction (i.e., $\frac{1}{2^n},n\in \mathbb{Z}$, e.g. $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32}...$) of it. Thus, we can measure any angle in terms of right angles. That is, a measurement system in which the size of the right angle is considered to be one, just as some believe that, with the distance measurement of feet, the length of King Henry I's foot was considered to be one.

Introduction to Radian Measure

Of course, it is equally possible to start with a different sized angle and split it into two equal angles. We might consider the angle made by half a circle to be "one", or the angle made by a full circle to be "one". Trigonometry is simplified if we choose the following strange angle as "one": Draw a circle with the radius marked. From the point where the radius meets the circumference of the circle, measure one radius length along the circumference of the circle moving counterclockwise and mark this point. Draw a line from this mark back to the center of the circle. The angle so formed is considered to be of size one in trigonometry; in order to tell your friends that this is the method you were using, you would say 'measuring in radians'.

Is a radian affected by the size of its circle?

Does it matter what size circle is used to measure in radians? Perhaps in large circles using one radius length along the circumference will produce a different angle than that produced by a small circle? When measuring out one radius length along the circumference of the circle we might proceed as follows:

• 1. Use a bit of string that is the length of one radius of the circle, and stick one end of the string to the starting point.
  2. Stick a small loop on the other end of the string and thread it along the circumference, moving it along the circumference until the string is pulled tight. The end of the string must reach beyond one cirumferal radian, because the string is now in a straight line as the circumference takes a longer path along the circle's edge.
  3. In order to make the string a better fit for the circumferance, find the half way point of the string, stick a loop there, pull it onto the circumference of the circle and pull the string (along the circle's circumferance) until the string is tight. The end point of the string is now closer to the one radian mark (along the circle's circumferance) because the string follows the circumference more closely, although still cutting across, this time in two sections.
  • 1. We can keep on improving the fit of the string against the circumference by dividing each new section of the string in half, sticking on a loop, threading the loop onto the circumference and pulling the string tight.

If we were to draw lines between neighboring loops where they touch the circumference and from each loop to the center of the circle, we would get a lot of isosceles triangles (triangles with two equal sides) whose longest sides are one circle's radius in length.

Now draw a small circle inside the first circle, with the same center, but with half the radius. It too has lots of isosceles triangles extending from its center to its circumference whose longest sides are one new circle radius (half of the original). The longest sides of the new isosceles triangles are half the size of the original isoceles triangles' matching sides. The old and new isosceles triangles share a common apex angle, and because they are isoceles, their other angles must also match. Therefore the old and new isosceles triangles are similar with half scale. Therefore the short edges of the new isosceles triangles are half the size of the old ones. Therefore the total length of the short sides of the new isosceles is close to half the length of the original radius, that is close to the length of the radius of the new circle. Hence the new triangles also delimit an angle as close to one radian as we like. Hence the definition of an angle of one radian is unaffected by the size of the circle used to define it.

Using Radians to Measure Angles

Once we have an angle of one radian, we can chop it up into binary fractions as we did with the right angle to get a vast range of known angles with which to measure unknown angles. A protractor is a device which uses this technique to measure angles approximately. To measure an angle with a protractor: place the marked center of the protractor on the corner of the angle to be measured, align the right hand zero radian line with one line of the angle, and read off where the other line of the angle crosses the edge of the protractor. A protractor is often transparent with angle lines drawn on it to help you measure angles made with short lines: this is allowed because angles do not depend on the length of the lines from which they are made.

If we agree to measure angles in radians, it would be useful to know the size of some easily defined angles. We could of course simply draw the angles and then measure them very accurately, though still approximately, with a protractor: however, we would then be doing physics, not mathematics.

The ratio of the length of the circumference of a circle to its radius is defined as 2π, where π is an invariant independent of the size of the circle by the argument above. Hence if we were to move 2π radii around the circumference of a circle from a given point on the circumference of that circle, we would arrive back at the starting point. We have to conclude that the size of the angle made by one circuit around the circumference of a circle is 2π radians. Likewise a half circuit around a circle would be π radians. Imagine folding a circle in half along an axis of symmetry: the resulting crease will be a diameter, a straight line through the center of the circle. Hence a straight line has an angle of size π radians.

The angles of a triangle add up to a half circle angle, that is, an angle of size π radians. To see this, draw any triangle, mark the midpoints of each side and connect them with lines to subdivide the original triangles into 4 similar copies at half scale. A copy of each of the original triangles three interior angles are juxtapositioned at each mid point, demonstrating that for any triangle the interior angles sum to a straight line, that is an angle of size π radians. This is true in particular for right angled triangles, which always have one angle of size π/2 radians, therefore the other two angles must also sum to a total size of π - π/2 = π/2 radians. If two sides of a right angled triangle have equal lengths, i.e. it is an isoceles right angled triangle, then each of the two other angles must be of size 1/2*π/2 = π/4 radians.

Folding a half circle in half again produces a quarter circle which must therefore have an angle of size π/2 radians. Is a quarter circle a right angle? To see that it is: draw a square whose corner points lie on the circumference of a circle. Draw the diagonal lines that connect opposing corners of the square, by symmetry they will pass through the center of the circle, to produce 4 similar triangles. Each such triangle is isoceles, and has an angle of size 2π/4 = π/2 radians where the two equal length sides meet at the center of the circle. Thus the other two angles of the triangles must be equal and sum to π/2 radians, that is each angle must be of size π/4 radians. We know that such a triangle is right angled, we must conclude that an angle of size π/2 radians is indeed a right angle.

Interior Angles of Regular Polygons

To demonstrate that a square can be drawn so that each of its four corners lies on the circumference of a single circle: Draw a square and then draw its diagonals, calling the point at which they cross the center of the square. The center of the square is (by symmetry) the same distance from each corner. Consequently a circle whose center is co-incident with the center of the square can be drawn through the corners of the square.

A similar argument can be used to find the interior angles of any regular polygon. Consider a polygon of n sides. It will have n corners, through which a circle can be drawn. Draw a line from each corner to the center of the circle so that n equal apex angled triangles meet at the center, each such triangle must have an apex angle of 2π/n radians. Each such triangle is isoceles, so its other angles are equal and sum to π - 2π/n radians, that is each other angle is (π - 2π/n)/2 radians. Each corner angle of the polygon is split in two to form one of these other angles, so each corner of the polygon has 2*(π - 2π/n)/2 radians, that is π - 2π/n radians.

This formula predicts that a square, where the number of sides n is 4, will have interior angles of π - 2π/4 = π - π/2 = π/2 radians, which agrees with the calculation above.

Likewise, an equilateral triangle with 3 equal sides will have interior angles of π - 2π/3 = π/3 radians.

A hexagon will have interior angles of π - 2π/6 = 4π/6 = 2π/3 radians which is twice that of an equilateral triangle: thus the hexagon is divided into equilateral triangles by the splitting process described above.

Summary and Extra Notes

In summary: it is possible to make deductions about the sizes of angles in certain special conditions using geometrical arguments. However, in general, geometry alone is not powerful enough to determine the size of unknown angles for any arbitrary triangle. To solve such problems we will need the help of trigonmetric functions.

In principle, all angles and trigonometric functions are defined on the unit circle. The term unit in mathematics applies to a single measure of any length. We will later apply the principles gleaned from unit measures to a larger (or smaller) scaled problems. All the functions we need can be derived from a triangle inscribed in the unit circle: it happens to be a right-angled triangle.

The hypotenuse rotate counter-clockwise

The center point of the unit circle will be set on a Cartesian plane, with the circle's centre at the origin of the plane — the point (0,0). Thus our circle will be divided into four sections, or quadrants.

Quadrants are always counted counter-clockwise, as is the default rotation of angular velocity $\omega$ (omega). Now we inscribe a triangle in the first quadrant (that is, where the x- and y-axes are assigned positive values) and let one leg of the angle be on the x-axis and the other be parallel to the y-axis. (Just look at the illustration for clarification). Now we let the hypotenuse (which is always 1, the radius of our unit circle) rotate counter-clockwise. You will notice that a new triangle is formed as we move into a new quadrant, not only because the sum of a triangle's angles cannot be beyond 180°, but also because there is no way on a two-dimensional plane to imagine otherwise.

Angle-values simplified

Imagine the angle to be nothing more than exactly the size of the triangle leg that resides on the x-axis (the cosine). So for any given triangle inscribed in the unit circle we would have an angle whose value is the distance of the triangle-leg on the x-axis. Although this would be possible in principle, it is much nicer to have a independent variable, let's call it phi, which does not change sign during the change from one quadrant into another and is easier to handle (that means it is not necessarily always a decimal number).

!!Notice that all sizes and therefore angles in the triangle are mutually directly proportional. So for instance if the x-leg of the triangle is short the y-leg gets long.

That is all nice and well, but how do we get the actual length then of the various legs of the triangle? By using translation tables, represented by a function (therefore arbitrary interpolation is possible) that can be composed by algorithms such as Taylor. Those translation-table-functions (sometimes referred to as LUT, Look up tables) are well known to everyone and are known as sine, cosine and so on. (Whereas of course all the abovementioned latter ones can easily be calculated by using the sine and cosine).

In fact in history when there weren't such nifty calculators available, printed sine and cosine tables had to be used, and for those who needed interpolated data of arbitrary accuracy - taylor was the choice of word.

So how can I apply my knowledge now to a circle of any scale. Just multiply the scaling coefficient with the result of the trigonometric function (which is referring to the unit circle).

And this is also why $\cos (\varphi {)}^2+\sin (\varphi {)}^2=1$, which is really nothing more than a veiled version of the pythagorean theorem: $\cos (\varphi )=a;\sin (\varphi )=b;a^2+b^2=c^2$, whereas the $c=1^2=1$, a peculiarity of most unit constructs. Now you also see why it is so comfortable to use all those mathematical unit-circles.

Another way to interprete an angle-value would be: A angle is nothing more than a translated 'directed'-length into which the information of the actual quadrant is packed and the applied type of trigonometric function along with its sign determines the axis ('direction'). Thus something like the translation of a (x,y)-tuple into polar coordinates is a piece of cake. However due to the fact that information such as the actual quadrant is 'translated' from the sign of x and y into the angular value (a multitude of 90) calculations such as for instance the division in polar-form isn't equal to the steps taken in the non-polar form.

Oh and watch out to set the right signs in regard to the number of quadrant in which your triangle is located. (But you'll figure that out easily by yourself).

I hope the magic behind angles and trigonometric functions has disappeared entirely by now, and will let you enjoy a more in-depth study with the text underneath as your personal tutor.

Next Page: Radians and Degrees
Previous Page: Introduction

Home: Trigonometry