Download - EdShare

Faculty of Mathematical Studies
MA101 Calculus - Outline Notes: Trigonometric Functions
You should know quite a lot about the three basic trigonometric functions - sin, cos, tan from your previous studies. There is a good deal of material in Adams, pages 41-53.
You may not have met the three reciprocal trigonometric functions:
Secant - the reciprocal of cosine.
Cosecant - the reciprocal of sine.
Cotangent - the reciprocal of tangent.
1
1
1
cos x
sec x 
; cosec x 
; cot x 

.
cos x
sin x
tan x sin x
The graphs of these three functions are shown on page 49. You don't need to learn them,
because you can reconstruct them from the graphs of cos, sin, tan - which you should know by taking reciprocals, as in some earlier examples.
Notice that cos x, sin x, sec x, cosec x all have period 2, whereas tan x, cot x have period .
Investigation What are the periods of cos 2 x, cos 3x,  cos kx ? What happens when you add
two of these, so what is the period of cos 5x  cos 7 x ? What is the period of cos mx  cos nx ?
What if m and n are not integers?
We can derive identities for these new functions from the ones we know for the three basic
functions, as in the following examples.
Example 1 Start with cos 2 x  sin 2 x  1.
Divide both sides by cos 2 x to give 1  tan 2 x  sec 2 x.
Divide both sides by sin 2 x to give cot 2 x  1  cosec 2 x.
Example 2 Using known addition formulae for cos and sin we can deduce that
cos( A  B) cos A cos B  sin A sin B cot A cot B  1
cot( A  B) 


.
sin( A  B) sin A cos B  cos A sin B cot B  cot A
(The last step is obtained by dividing numerator and denominator of the previous expression
by sin Asin B. )
The most important thing to realise about trigonometrical formulae is that many of them can
be deduced from a few basic ones.
Example 3 We shall make some deductions from the four addition formulae
1. cos( x  y )  cos x cos y  sin x sin y
2. sin( x  y )  sin x cos y  cos x sin y
3. cos( x  y )  cos x cos y  sin x sin y
4. sin( x  y )  sin x cos y  cos x sin y
In fact if we know that cos is an even function and sin is an odd function we can deduce 3.
from 1. and 4. from 2. by replacing x by x.
If we know that cos 0  1 then putting y  x in 3. gives cos 2 x  sin 2 x  1.
On the other hand if we were to assume that cos 2 x  sin 2 x  1 then putting y  x in 3.
would give cos 0  1. This serves to emphasise that there is a network of relationships
between the various trigonometric formulae.
Putting y  x in 1. and 2. gives the double angle formulae
cos 2 x  cos 2 x  sin 2 x
sin 2 x  2 sin x cos x
We can then find formulae for cos 3x for example by using cos 3x  cos( 2 x  x) together
with the addition formula and the double angle formulae.
Some identities which are useful in integrating products of trigonometric functions can be
obtained by adding the addition formulae.
For example adding 1. and 3. Gives 2 cos x cos y  cos( x  y )  cos( x  y ), and so we can
convert a product into a sum of trigonometric functions, which is generally easier to integrate.
You should find for yourself similar formulae for sin x sin y and sin x cos y.