WHY LEARN TRIGONOMETRY?
The best reason that I have found was given by Thomas Paine:
In Chapter XI of The Age of Reason, the American revolutionary and Enlightenment thinker Thomas Paine wrote:
The scientific principles that man employs to obtain the foreknowledge of an eclipse, or of any thing else relating to the motion of the heavenly bodies, are contained chiefly in that part of science that is called trigonometry, or the properties of a triangle, which, when applied to the study of the heavenly bodies, is called astronomy; when applied to direct the course of a ship on the ocean, it is called navigation; when applied to the construction of figures drawn by a ruler and compass, it is called geometry; when applied to the construction of plans of edifices, it is called architecture; when applied to the measurement of any portion of the surface of the earth, it is called land-surveying. In fine, it is the soul of science. It is an eternal truth: it contains the mathematical demonstration of which man speaks, and the extent of its uses are unknown.
MAKING THE UNDERSTANDING OF TRIGONOMETRY EASIER:
When learning trigonometry, I found that the representation of a rotating unit vector was very helpful in visualizing sine and cosine values.
Please take a look at the following articles to provide a background for what follows:
https://en.wikipedia.org/wiki/Trigonometric_functions
https://en.wikipedia.org/wiki/Unit_vector
The sine value is defined as the ratio of the length of the side opposite the angle in question, to the length of the hypotenuse; in a right (90 degrees) triangle. A hypotenuse length of 1, simplifies things considerably.
The cosine value is defined as the ratio of the length of the side adjacent the angle in question, to the length of the hypotenuse; in a right (90 degrees) triangle.
The tangent value is defined as the ratio of the sine value to the cosine value.
When working with the unit (length of 1) vector, ratios become easier because the sine is now directly equal to the side opposite length and the cosine to the side adjacent length.
REPRESENTING AN ANGLE BY THE X AND Y AXIS PROJECTIONS:
I discovered the Trammel of Archimedes 3D toy by Igbu on the website ‘thingiverse’. I was reminded of the fact that sine and cosine values are simply the projection (values) of the unit vector on the x (cosine) and y (sine) axes. I created a 3D model from this toy to show trigonometric relationships. Please see:
https://en.wikipedia.org/wiki/Trammel_of_Archimedes
I modified the design in ‘TinkerCad’ to produce this prototype:
See ‘Thingiverse.com’ and search for GeeEaZy for an opportunity to 3D print the finished model.
To use; you rotate the pointing bar to the angle in question. In the picture above, the angle selected is approximately +45 degrees. The Sine value is read from the y axis; the red slider. The Cosine value is read from the x axis; the green slider. The values range from 1 to -1. For the Sine values +1 is at the top (90 degrees) and -1 at the bottom (270 degrees). 0 is in the middle. Cosine values are +1 on the left (180 degrees) and -1 on the right (0 degrees). 0 is in the middle.
HERE ARE SOME EXAMPLES:
Place the pointer at 0 degrees and read the Sine value (red) on the y axis as 0 and the cosine value (green) on the x axis as 1.
Place the pointer at 90 degrees and read Sine value (red) as 1 and cosine value (green) as 0.
Place the pointer at 180 degrees and read Sine value (red) as 0 and cosine value (green) as -1.
Place the pointer at 270 degrees and read Sine value (red) as -1 and cosine value (green) as 0.
Angles in between must be interpolated to give approximations of the values.
HERE IS AN ENHANCED and final VERSION:
I found that the model also serves in aiding visualization of the tangent value. The tangent value is defined as sine value divided by the cosine value, which can be visualized as a distance on the tangent line. Visualize the distance starting where the tangent intersects the unit vector at 90 degrees, and extends to where the tangent line of the selected angle intersects the x axis. For example: Set the pointer at angle 0, the distance starts and ends at this point, so is 0. Now set the pointer to the desired angle, sight along the tangent line to where it intersects the x axis. The tangent value is approximately that distance.
Please note! That at about 89 degrees (a smidge short of 90) the tangent line is almost parallel to the x axis and the tangent value is quite large in the +x direction. At 90 degrees it becomes infinite, never intersecting the x axis. At about 91 degrees (a smidge past 90) it becomes almost infinite in the other polarity (-x direction). Quite a tangent value change resulting from a small angle degree change! Look at the discontinuous graphs of the tangent value around 90 degrees. Quite strange until you see it with the model!
I have found a number of memory aids for trigonometry, but believe this to be a unique approach. Thanks to lgbu and to my High School Trigonometry teacher who wouldn’t stop drawing blackboard diagrams until we understood it.