Trigonometry/For Enthusiasts/Trigonometry Done Rigorously
|This page started life as an introduction to the most basic concepts of trigonometry, such as measuring an angle. Done properly this is an advanced topic. This page has now been moved/renamed into book 3 to take on that role, and needs considerable revision. Some content that is still here may need to be integrated back into book 1|
Introduction to Angles
An angle is formed when two lines intersect; the point of intersection is called the vertex. We can think of an angle as the wedge-shaped space between the lines where they meet. Note that if both lines are extended through the meeing point, there are in fact four angles.
The size of the angle is the degree of rotation between the lines. The more we must rotate one line to meet the other, the larger the angle is.
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.
Definition of an Angle
An angle is determined by rotating a ray about its endpoint. The starting position of the ray is called the initial side of the angle. The ending position of the ray is called the terminal side. The endpoint of the ray is called its vertex. Positive angles are generated by counter-clockwise rotation. Negative angles are generated by clockwise rotation.
Consequently an angle has four parts: its vertex, its initial side, its terminal side, and its rotation.
An angle is said to be in standard position when it is drawn in a cartesian coordinate system in such a way that its vertex is at the origin and its initial side is the positive x-axis.
Definition of a Triangle
A triangle is a planar (flat) shape with three straight sides. An angle is formed between each two sides of a triangle, and a triangle has three angles, hence the name tri-angle. So a triangle has three straight sides and three angles.
If you give me three lengths, I can only make a triangle from them if the greatest length is less than the sum of the other two. Three lengths that do not make the sides of a triangle are your height, the height of the nearest tree, the distance from the top of the tree to the center of the sun.
Angles are not affected by the length of lines: an angle is invariant under transformations of scale, that is:
An angle of particular significance is the right angle: the angle at each corner of a square or a rectangle. A rectangle can always be divided into two triangles by drawing a line from one corner of the rectangle to the opposite corner.
It is also true that every right-angled triangle is half a rectangle.
A rectangle has four sides; they are generally of two different lengths: two long sides and two short sides. (A rectangle with all sides equal is a square.) When we split the rectangle into two 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.
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 complete 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
It is possible to bisect any angle using only circles (which can be drawn with a compass) and straight lines by the following procedure:
- Call the vertex of the angle O. Draw a circle centered at O.
- Mark where the circle intersects each ray. Call these points A and B.
- 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.
- 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., , e.g. ) 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.
Introduction to Radian Measure
Trigonometry is simplified if we choose the following strange angle as "one":
Understanding the three sides of a Radian
To illustrate how the three sides of a radian relate to one another try the below thought experiment:
- Assume you have a piece of string that is exactly the length of the radius of a circle.
- Assume you have drawn a radian in the same circle. The radian has 3 points. One is the center of your circle and the other two are on the circumference of the circle where the sides of the radian intersect with the circle.
- Attach one end of the string at one of the points where the radian intersects with the circumference of the circle.
- Take the other end of the string and, starting at the point you chose in Step 2 above, trace the circumference of the circle towards the second point where the radian intersects with the circle until the string is pulled tight.
- You will see that the end of the string travels past the second point.
- This is because the string is now in a straight line. However, the radian has an arc for its third side, not a straight line. Even though a radian has three equal sides, the arc's curve causes the two points where the radian intersects with the circle to be closer together than they are to the third point of the triangle, which is at the center of our circle in this example.
- Now, with the string still pulled tight, find the half way point of the string, then pull it onto the circumference of the circle while allowing the end of the string to move along the circle's circumference. The end of the string is now closer to the second point because the path of the string is closer to the path of the circle's circumference.
- We can keep improving the fit of the string's path to the path of the circle's circumference by dividing each new section of the string in half and pulling it towards the circumference of the circle.
Is a radian affected by the size of its circle?
Does it matter what size circle is used to measure in radians? Perhaps using the radius of a large circle will produce a different angle than that produced by the radius of a small circle. The answer is no.
Recall our radian and circle from the experiment in the subsection above. Draw another circle inside the first circle, with the same center, but with half the radius. You will see that you have created a new radian inside the smaller circle that shares the same angle as the radian in the larger circle. We know that the two sides of the radians emanating from the center of the circles are equal to the radius of their respective circle. We also know that the third side of the radian in the larger circle (the arc) is also equal to the larger circle's radius. But how do we know that the third side of the radian in the smaller circle (the arc that follows the circumference of the smaller circle) is equal to the radius of the smaller circle? To see why we do know that the third side of the smaller circle's radian is equal to its radius, we first connect the two points of each radian that intersect with the circle with each other. By doing so, you will have created two isosceles triangles (triangles with two equal sides and two equal angles).
An isosceles triangle has two equal angles and two equal sides. If you know one angle of any isosceles triangle and the length of two sides that make up that angle, then you can easily deduce the remaining characteristics of the isosceles triangle. For instance, if the two equidistant sides of an isosceles triangle intersect to form an angle that is equal to 40o, then you know the remaining angles must both equal 70º. Since we know that the equidistant sides of our two isosceles triangles make up our known angle, then we can deduce that both of our radians (when converted to isosceles triangles with straight lines) have identical second and third angles. We also know that triangles with identical angles, regardless of their size, will have the length of their sides in a constant ratio to each other. For instance, we can deduce that an isosceles triangle will have sides that measure 2 meters by 4 meters by 4 meters if we know that an isoscleses triangle with identical angles measures 1 meter by 2 meters by 2 meters. Therefore, in our example, our isosceles triangle formed by the second smaller circle will have a third side exaclty equal to half of the third side of the isosceles triangle created by the larger circle. The relationship between the size of the sides of two triangles that share identical angles is also found in the relationship between radius and circumference of two circles that share the same center point - they will share the exact same ratio. In our example then, since the radius of our second smaller circle is exactly half of the larger circle, the circumference between the two points where the radian of the smaller circle intersect (which we have shown is one half of the distance between the two similar points on the larger circle) will share the same exact ratio. And there you have it - the size of the circle does not matter.
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.
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.
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 trigonometric 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 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 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). 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.