Friday Gen Eds MS9: Basic Geometry and Trigonometry

This is the ninth post in the math/science part of my "Gen Eds in a Nutshell" series. It's a series of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm over half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:

• Math/Science Overview
• Basic Types of Numbers
• The Atom and Quantum Physics
• The Periodic Table
• Molecules and Ions
• Chemical Reactions
• The Basic Tools of Algebra
• Heat and Thermodynamics

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1. From a popular perspective, geometry is the study of shapes and spaces. More properly, it is the mathematical study of points, lines, planes, solids, and higher dimensions by analogy.

A point is a placeholder for a single location in space, a "dimensionless" entity. Two points make a line, or at least we can draw one and only line through those two points (at least in the geometry we learn in the ninth or tenth grade). A line is a "one-dimensional" entity. Three points identify a plane, a "two-dimensional" concept. Solids are thus three-dimensional entities.

2. The basic shapes have been known and used by humanity from its earliest memories--circles, triangles, squares. The areas of these shapes are the amount of space they take up. The area of a rectangle is easy enough to calculate. it is the measure of one side times the measure of another. The very idea of squaring (x²) comes from the fact that the area of a square is the measure of one side times itself.

The Babylonians, Egyptians, and Greeks knew that the ratio of the circumference of a circle divided by its diameter was a constant, and Archimedes estimated it at around 3.14. But it wasn't until the 1700s that the symbol π was used (for perimeter, since π is the perimeter of a circle). The area of a circle is equal to πr², were r is the radius of the circle.

3. Since the Babylonians and perhaps even before, it has been to divide up a circle into 360 degrees. This is completely arbitrary from our standpoint. It's much more helpful to measure a circle in terms of pi, where the perimeter of a circle is 2π. This is called radian measurement. One radian is the measure of an arc of the circle equal to the measure of the radius.

But we are still used to measuring a circle in terms of 360 degrees. We are used to measuring the angles of a triangle in these degrees as well. A triangle has three angles. The total of these three angles always adds up to 180 degrees, or half a circle. If you were to measure the arc in a circle from one side of a line to the other, you would find that it also covers 180 degrees. That's why we say we turned 180 degrees, meaning that we completely changed directions.

There are special kinds of triangles. For example, an equilateral triangle is one where all three sides have the same measure. This means that all three angles also have the same measure. Since the total is 180 degrees, each angle of an equilateral triangle must measure 60 degrees. An isosceles triangle is one where two sides have the same measure. That means that two of the angles, the ones opposite the sides with similar measures, will have the same measure as well.

A right triangle is one where one of the angles of the triangle is 90 degrees. The other two angles then add up to the remaining 90. Pythagoras in the 500s BC knew the relationship between the measures of the sides of a right triangle: a² + b² = c². This formula is called the Pythagorean theorem.

4. Geometry is usually the subject where students learn about proofs. Modern mathematics is based on a structure of axioms and theorems. Axioms or postulates are assumptions that cannot strictly be proved. They are like definitions or starting points. "For any two points, there is one and only one line that goes through them."

Axioms are usually commonsensical assumptions like "Two parallel lines never meet." However, every once and a while, what seems obvious may not be true. Since the 1700s, a whole system of geometry has been developed from removing this assumption. Euclidean geometry is geometry that assumes this "parallel postulate" as the ancient Greek Euclid (ca. 300BC) did.

By contrast, non-Euclidean geometry explores mathematics without the assumption of the parallel postulate. As it turns out, this form of geometry has been very helpful with the theory of relativity. Our common sense assumptions have turned out not to be as useful for science on this level as we thought.

Proofs then take these axioms and build theorems out of them using some basic rules of logic. For example. The "transitive property of equality" states that "if a = b and b = c, then a = c." These are the sorts of rules are used to glue axioms together into theorems.

5. Some of these theorems show when two triangles are "congruent." For example, the "side-side-side" (SSS) theorem states that if the three sides of two triangles have the same measures, then the two triangles are equivalent or congruent. There are also "side-angle-side" (SAS), "angle-side-angle" (ASA), and "angle-angle-side" (AAS) theorems.

There are also a set of theorems relating to the angles around two parallel lines. For example, if a line cuts across two parallel lines, then the "alternate interior angles" are congruent.

6. We know rectangles where all the angles of a four sided figure are 90 degree angles. A square is thus a special kind of rectangle where the four sides all have the same measure.

Any four sided figure is called a quadrilateral. If the opposite sides are parallel to each other, it is called a parallelogram. A square and rectangle is thus a special kind of parallelogram where the opposite sides are not only parallel, but at 90 degrees to each other.

A trapezoid is a quadrilateral where only one pair of opposite sides are parallel. A rhombus is a parallelogram where all four sides have an equal measure.


7. We can conceptualize another set of shapes as conic sections. If you look at the inverted cones to the right, you can create several shapes by taking a cross-section of it. A straight horizontal slice cuts out a circle, while a slanted slice through a cone gives an ellipse, an elongated circle.

If you slice slanted but in a way that does not cut all the way through, then you have a parabola. Finally, if you cut up and down vertically, you get the equivalent of two opposite parabolas, which is called a hyperbola.

8. These sorts of explorations took a massive leap forward when René Descartes developed his Cartesian coordinate system (see post on basic algebra). The branch of mathematics known as analytic geometry looks at geometry from a graphical standpoint. A point is thus not just an abstract concept, but something that can be plotted on a graph. If we have a two dimensional graph with an x-axis and a y-axis, we might locate a point with the format (x, y), where the first number gives us where the point is on the horizontal, and the y tells us where the point is on the vertical.

Our minds can also conceptualize a three dimensional framwork with x, y, and z axes. Points on this sort of a graph are plotted by three points in the format (x, y, z).

It is difficult for us to imagine a graph for more than three dimensions. Usually, additional dimensions are plotted in two dimensions. For example, if you want to plot movement in space in relation to time, you would put "t" as an axis over and against a distance axis (see next post). Imaginary numbers are sometimes graphed in terms of an imaginary axis over and against a real axis (see post on basic types of numbers).

9. We already discussed the basic graph of a line: y = mx + b. In this version of the equation (slope-intercept form), m is the slope of the line and b is the y-intercept point. The conic sections we mentioned above also have basic forms:

• A circle can be put into the form of (x - h)² + (y - k)² = r², where (h, k) is the center point and r is the radius. x and y are then the points on the circle.



Formula for an ellipse

• An ellipse can be put into the form to the right. In this formula, (h, k) is the center of the ellipse and a and b are the distances in perpendicular directions from the center to the ellipse itself.
• A quadratic equation, an equation with a square as its highest power, usually graphs as a parabola. The standard equation format for a parabola is y=a(x - h)² - k, where (h, k) is the vertex point (the h and k can be switched depending on which way the parabola is facing) and a tells us how squeezed the parabola is.

• Finally, a hyperbola takes the form to the right, where (h, k) is the point at the center point between the two "parabolas" of the hyperbola, a represents the distance from this point to the vertices of the two branches of the hyperbola and b is the distance from the center point to where the "asymptotes" that box in the parabola meet a line going through the vertices of each wing. [1]

10. Trigonometry is the geometry of right triangles. In particular, there are certain relationships between the measures of the sides of a right triangle and the angles of that triangle:

• The "sine" (sin) of an angle is the ratio of the opposite side divided by the hypotenuse (the side opposite the right angle).
• The "cosine" (cos) of an angle is the ratio of the side adjacent to the angel divided by the hypotenuse.
• The "tangent" (tan) of an angle is the ration of the side opposite an angle divided by the side adjacent.
• There are "reciprocals" to these, basically 1/them. The reciprocal of the tangent is the cotangent (adjacent/opposite side measure). The reciprocal of the cosine is the secant (hypotenuse/adjacent) and of the sine is the cosecant (hypotenuse/opposite).
• You can also find the angle by taking the arcsine, the arccosine, and the arctangent of the sine, cosine, and tangent respectively. This words backwards to the angle. So if you take the sine of the angle, you get the ratio between the opposite side and the hypotenuse. If you take the inverse, the arcsine of the ratio, you get the measure of the angle.

11. We might now mention that volume has to do with the three-dimensional space occupied by a three-dimensional figure. So the idea of cubing a number (x³) comes from finding the volume of a cube, which is a three-dimensional square of sorts, a box with all sides equal. The volume is the length times the width times the height.

For a sphere, a three-dimensional circle of sorts or globe, the volume is found by the formula 4/3 πr³. Surface area refers to the area of the space on the surface of a three-dimensional figure. For a cube, for example, there are 6 sides, each of which has the area of a square. So the surface area of a cube is 6x². Meanwhile, the surface area of a sphere is 4πr².

12. I mentioned above the rise of non-Euclidean geometry above. Hyperbolic or Lobachevskian geometry is an example of such geometry. The key is not to think of the space on which the geometry of lines and shapes is playing out as a flat space. So in hyperbolic geometry, we might think of geometry playing itself out on a saddle that bends inward. For example, a triangle drawn on a saddle will have angles that add up to something less than 180 degrees.

The opposite is spherical geometry, where we might think of geometry playing itself out on a sphere. In this case, the angles of a triangle drawn on a sphere will add up to more than 180 degrees.

Next Week: Math/Science 10: The Physics of Motion

[1] An asymptote is a line that goes onto infinity that serves as a kind of imaginary line that a graph is not allowed to cross. Take the equation y = 1/x (the inverse function). Since you cannot divide by 0, x cannot ever equal 0 for this equation. If you graph it