The Science and Math Connection

Students learn most effectively if they are able to apply inquiry and problem-solving skills to problems that emphasize practical applications. Many experts stress connecting science to other disciplines, such as mathematics, and modeling word problems to real-world situations.

Making a connection between mathematics and chemistry (determining the optimal angle between the atoms of covalent bonds) should help answer the trigonometry student's question, "Why do we need to learn these identities and when will we ever use them?" Several trigonometric identities are necessary when doing the proof of the optimal angle for a molecule with four identical atoms bonded to a central atom that has a complete valence shell.

BASIC INFORMATION

Atoms form covalent bonds with other atoms to create molecules. A covalent bond is formed when two atoms share a pair of electrons.

POLYATOMIC MODELING

A more interesting problem of molecular geometry is encountered when dealing with a molecule comprised of four atoms covalently bonded to a central atom leaving no nonbonding electron pairs. A common example of such a molecule is methane (CH₄). The Lewis structure for CH₄ also appears in Figure 1. The Lewis structure suggests that the optimal bond angle for methane is 90°. Does a three-dimensional conformation exist for methane that would allow bond angles greater than 90°? If such a conformation exists, the hydrogen atoms would be farther apart from each other. How does one go about finding the optimal bond angle that places these four hydrogen atoms at points in space that are the greatest distance from each other?

THREE-DIMENSIONAL MODELS

The three-dimensional model of methane is a tetrahedron, with the carbon atom at the center of the tetrahedron and the four hydrogen atoms at the vertices (Figure 3).

To determine the optimal bond angle, draw a perpendicular line from the carbon atom (C) to the plane containing three of the hydrogen atoms.
Let Q represent the foot of this perpendicular line (Figures 3 and 4), and
let y represent the distance between the carbon atom and any of the hydrogen atoms.
Let a represent the distance from Q to one of the hydrogen atoms, and
let x represent the measure of the required bond angle.

In Figure 3, note that Q is the circumcenter of the equilateral triangle formed by the three hydrogen atoms that lie in the bottom plane. Because the triangle HHH is equilateral, each of the angles HQH measures 120°. In Figure 4, the measure of angle HCQ = (180-x)° and the measure of angle CHQ = (x-90)°.

Statements	Reasons
	Definition of cosine = adjacent / hypotenuse
a = y cos(x- 90)°	Multiplication Property of Equality
a = y (cos(-(90-x)))	Distributive Property (Factored out -1)
a = y (cos(90 - x))	cos(-A) = cos(A)
a = y sin(x)	cos (90 - A) = sin(A)

Now, examine the triangle formed by two hydrogen atoms and point Q (Figure 5). The altitude from point Q divides the triangle HQH into two congruent triangles HQT and HQT (hypotenuse - leg theorem). So, the vertex angle HQH is divided into two angles whose measures are each 60°.

Figure 6 represents the triangle formed by the carbon atom and two hydrogen atoms.

Statements	Reasons
	Definition of sine = opposite / hypotenuse
	Substitution (Sub a = y sin(x) into previous step)
	Square both sides
2 - 2cos(x) = 3 (1 - cos2x)	Multiplication Property of Equality
2 - 2cos(x) = 3 - 3 cos2x	Distributive Property
3 cos2x - 2 cos(x) - 1 = 0	Addition Property of Equality
(3 cos(x) + 1) (cos(x) - 1) = 0	Factor (Distributive Property)
cos (x) = -1/3 or cos(x) = 1	Set each factor = 0 (Mult. Prop. of 0) Muliplication Property of Equality
so x = 109.4712206°	Inverse Cosine

CONNECTING MATH TO SCIENCE

It is important for students to make connections between mathematics and other disciplines. Knowledge of mathematics means much more than just memorizing information or facts; it requires the ability to use information to reason, think, and solve problems. By themselves, trigonometric identities are just facts, but applying them to a real-world problem will give students a deeper appreciation of those identities and of mathematics. This manipulation of several trigonometric identities allows students to discover for themselves that the optimal bond angle for methane is 109.5°, not 90° as suggested by the 2-dimensional representation. Hopefully, students will begin to value and use the connections between mathematics and other disciplines.