2023 ASEE Annual Conference & Exposition

Making engineering education accessible to under prepared students entering college from high school and students transitioning from the community college level is sometimes difficult due to the demanding mathematical requirements the major demands. One specific area of great difficulty for under prepared students is understanding the trigonometric and inverse trigonometric functions. Part of the problem is that the trigonometric functions seem mysterious to them because they are only seen as buttons on a calculator. The trigonometric functions are classified as transcendental functions. A transcendental function cannot be written as a finite combination of algebraic expressions. The key word is FINITE. This fact in most cases eliminates the equation form from ever being seen by students. Students know them as only a word sine, cosine, and tangent that is somehow related to the sides of a right triangle. Below are the actual formulas for sine, cosine, and tangent functions. For simplicity in computational purposes only the first three terms in the series will be used. Using these formulas will give under prepared incoming engineering students the hands on feel of working with familiar functions such as y = f(x) = 3x2 + 2x - 4. They are familiar with the independent variable x and the dependent variable y. This paper is intended to help under prepared students understand the trigonometric functions and the notation used to represent them. Most students don’t realize that the f in f(x) is being replaced by sin, cos, and tan. It will then be explained that these formulas are programmed into their calculators and are accessible by the sin, cos, and tan buttons on a calculator.
b/c = f(x) = sin(x) = x – x3/6 + x5/120 b/a = f(x) = tan(x) =x + x3/3 + 2x5/15
a/c = f(x) = cos(x) = 1 – x2/2 + x4/24 c2 = a2 + b2
This paper is not written from a research perspective. There was no collected student data. This paper will contain a full written abbreviated chapter that can be included in any first semester trigonometry or physics course. Formula derivations will not be included, and knowledge of radian measure will be assumed. It will contain several fully worked example problems. The problems will contain the use of the above functions where students only use a calculator to calculate the first three terms given in the above formulas. It is intended only as a learning resource and can be used by any math or engineering educators.
The example problems will emphasize the use of the right triangle ratios b/c, a/c, and b/a as the dependent variable. Graphs of sine and cosine will also be given with the vertical axis labeled as b/c and a/c as opposed to y. Some students have difficulty understanding why the right triangle ratios simply disappear when graphing the trigonometric functions. This supplemental chapter will hopefully reinforce the idea that the trigonometric functions require a real number input.