Mathematics for chemistry/Trigonometry

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Free Web Based Material from UK HEFCE

There is a DVD on trigonometry here[1].

Trigonometry - the Sin and Cosine Rules

        The sin rule

           a                        b                 c
          ----         =          ----      =       ----
          sinA                    sinB              sinC


        The Cosine Rule



            2         2           2
           c    =    a     +     b         - 2 a b cosC



                    sinA
           tanA  =  ----
                    cosA

The Trigonometric Identities

${\cos }^2\theta +{\sin }^2\theta =1$

Remember this is a consequence of Pythagoras' theorem where the length of the hypotenuse is 1. $\sin (\theta +\varphi )=\sin \theta \cos \varphi +\cos \theta \sin \varphi$

$\cos (\theta +\varphi )=\cos \theta \cos \varphi -\sin \theta \sin \varphi$

The difference of two angles can easily be generated by putting $\varphi =-\varphi$ and remembering $\sin -\varphi =-\sin \varphi$ and

$\cos -\varphi =\cos \varphi$.

Similarly the double angle formulae are generated by induction. $\tan (\theta +\varphi )$ is a little more complicated but can be generated if you can handle the fractions!

The proofs are in many textbooks but as a chemist you do not need to know how to do this but you do need the results.

Identities and Equations

Identities and equations look very similar, two things connected by an equals sign. An indentity however is a memory aid of a mathematical equivalence and can be proved. An equation represents new information about a situation and can be solved.

For instance: ${\cos }^2\theta +{\sin }^2\theta =1$ is an identity. It cannot be solved for $\theta$. It is valid for all $\theta$. However: ${\cos }^2\theta =1$ is an equation where $\theta =\arccos \pm 1$.

If you try and solve an identity as an equation you will go round and round in circles getting nowhere, but it would be possible to dress up ${\cos }^2\theta +{\sin }^2\theta =1$ into a very complicated expression which you could mistake for an equation.

Some Observations on Triangles

Check you are familiar with your elementary geometry. Remember from your GCSE maths the properties of equilateral and iscoceles triangles. If you have an iscoceles triangle you can always dispense with the sin and cosine rules, drop a perpendicular down to the base and use trig directly. Remember that the bisector of a side or an angle to a vertex cuts the triangle in two by area, angle and length. This can be demonstrated by drawing an obtuse triangle and seeing that the areas are $\frac{1}{2}\frac{1}{2}.R.h$.

The Interior Angles of Polygon

Remember that the interior angles of a $n$-sided polygon are n * 180 -360,

($n\pi -2\pi$).

For benzene we have 6 equilateral triangles if we use the centre of the ring as a vertex and have an interior angle of 120 degrees. Work out the the angles in azulene, (a hydrocarbon with a five and a seven membered ring), assuming all the C-C bond lengths are equal, (this is only approximately true).

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