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Trigonometric functions of half argument
The formulas for the Trigonometric functions of half argument are:
, ,
,
, , .
Three formulas in the last line above allow to represent/to express the major trigonometric functions , and as the rational functions of .
This implies that any rational function of , and can be expressed as a rational function of only.
In this lesson you can learn how to prove these formulas.
Proof of the half argument formula for cosines
We are going to prove the formula
,
which is the same as the formula
.
The proof is very simple and straightforward.
It is based on the addition formula for cosines of the lesson Addition and subtraction formulas in this module:
.
Simply take , in this formula. You get
.
Now, substitute here
.
You get
.
The last step is to make a simple rearrangement in this formula
,
which is exactly what we are going to prove.
The proof is completed.
The formula
.
is the immediate consequence of the previous one.
Proof of the half argument formula for sines
We are going to prove the formula
which is the same as the formula
.
In the section above we just proved that
.
Therefore,
,
which is exactly what we are going to prove.
The proof is completed.
The formula
.
is the immediate consequence of the previous one.
Proof of the half argument formula for tangents
We are going to prove the formula
.
We have
(after multiplying the numerator and denominator by )
= (after applying the half argument formula for cosines proved above in this lesson).
This is exactly what we are going to prove.
Regarding the formula
,
the proof is very similar. We have
(after multiplying the numerator and denominator by )
= (after applying the half argument formula for sines proved above in this lesson).
This is exactly what we are going to prove.
The proof is completed.
For examples of the applications of these formulas see the lesson Trigonometric functions of half argument - Examples in this module.
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For your convenience, below is the list of my lessons on Trigonometry in this site in the logical order.
They all are under the current topic Trigonometry in the section Algebra II.
Addition and subtraction formulas
Addition and subtraction of trigonometric functions
Product of trigonometric functions
Powers of trigonometric functions
Trigonometric functions of multiply argument
Trigonometric functions of half argument
Miscellaneous Trigonometry problems
The lesson Miscellaneous Trigonometry problems
Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.
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