6. Trigonometry
Radians or degrees
The calculus of trigonometric functions involves radians, and they are often the unit of choice in mathematics. Always assume that an angular quantity is in radians (unless there is something suggesting otherwise [a degree symbol for example]). All identities will still work with radians as radians are simply another way of measuring the angle.
Secant, cosecant and cotangent
There are three more trigonometric functions that are in common use, these are simply the reciprocals of the already known cosine, sine, and tangent. Hence:
(Note that 
also).
The standard trigonometric function keys on a calculator coupled with the reciprocal key will enable the calculation of these functions, as generally there are no buttons specifically for these functions.
The graphs of these functions are easy enough to predict. Consider that sine and cosine fluctuate between -1, and 1; this means that secant, and cosecant will never have a value whose modulus is below 1, and will go off to infinity (in both directions). Also, the period of cosecant, and secant are the same as their corresponding functions (sine, and cosine).
These new functions can be used in the context of identities, and the Pythagorean identity is a good starting point.
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These identities can be used to prove further identities, and hence to solve equations.
Example
1. Prove that 
.
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The addition formulae for sine and cosine
It is possible (with the correct knowledge) to calculate the sine, or cosine of a sum of two angles:
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To derive the formula it is helpful to consider two points, R, and S (
, and 
respectively).
The distance between these two points is:
This distance corresponds to a side of a triangle which has an angle (at the origin) of (A - B).
If we consider the unit circle as a definition of where these points are, it is possible to calculate this distance in terms of the above angle (using the cosine rule).
Hence:
Hence (through the use of the Pythagorean identity):
.
Similarly, for sine, 
.
Hence:
.
(Note that the identity of 
was used in the above).
It is therefore possible to construct the following table of addition (and subtraction) forumlae:
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The addition formulae for tangents
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This can be used to formulate an addition formula for tangent.
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Double angle formulae
The double angle formulae are simply special cases of the addition formulae, in which there is a doubling involved.
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The double angle formula for the cosine is interesting as the Pythagorean identity can be applied directly to it (yielding helpful [and therefore noteworthy] results):
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These formulae are heavily used in proof of identity, and can also be used in trigonometric equations (in order to change the equation into one that is easier to solve).
The form 
Functions in the form of 
can be represented as a single trigonometric function (if the function was graphed it would appear to be a sine, or cosine curve which had been translated and stretched).
The idea behind this is the addition formulae. Consider that is that same as 
.
At this point the equivalences are what is important. As one statement is equivalent to the other, the variable ("x") can be any value, and there is equality, so allowing 
, or 
can remove one or other of the unknowns.
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As 
:
.
Once "R" is known, the value of "y" can be calculated, and the statement can be completed.
In general:
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a \sin{x} + b \cos{x} \equiv \sqrt{ a^{2} + b^{2} } \left( \sin{ x + \sin ^{-1} { \left( \frac{b}{ \sqrt{ a^{2} + b^{2} } \right) } } \right)
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Notice that this is a transformation of the sine curve. A stretch in the y-direction, scale factor
, followed by a translation in the x-direction of
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- \sin ^{-1} { \left( \frac{b}{ \sqrt{ a^{2} + b^{2} } \right)}
units.
Do not learn this formula, learn the method. Other forms are better suited to other trigonometric functions:
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This representation of a function can be used to solve equations involving both sine and cosine.
Inverse trigonometric functions
The inverse trigonometric functions will have been used heavily during the solution of trigonometric equations, however, without the knowledge of functions gained in Core 3 it is unlikely that they will have been thought of as inverse functions, and issues such as domain and range will not have been consciously considered.
As sine, cosine, and tangent are not one-one they must have a restriction on their domain such that all values in the range are represented, but they can have a defined inverse. The implication of this is that the inverse trigonometric functions will always output a value within a specific range (the restricted domain of the trigonometric functions).
For cosine, restricting the domain to 
is sufficient to encompass all values in the range 
, and there are no values for which there is more than one value in the domain that would produce that value when the function is applied to it.
Sine is different. Consider the identity: 
. This implies that if the same domain restrictions as cosine were applied there would be problems (as the function would not be one-one). Due to the asymmetry about the y-axis of the sine function it is possible to have a domain of 
. In this domain all values in the range of the sine function can be produced, and the function is also one-one.
Tangent is similar to sine, and can be restricted using the same domain.
Inverse functions are reflections of the original functions in the line 
; this is a property which allows the visualisation of the graphs of the inverse trigonometric functions. After restricting the domains of the trigonometric functions the x- and y-axis can swap around, and the graph can be rotated such that the x-axis runs at the bottom (for convention); these graphs are now the graphs of the inverse trigonometric functions. Note that a value of "x" will produce a value that corresponds to the angle whose sine, cosine, or tangent (depending on the graph being considered) is that value ("x").
