Lesson Trigonometric functions of half argument

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Trigonometric functions of half argument


The formulas for the Trigonometric functions of half argument are:

sin%5E2%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2F2, cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2,

tan%28alpha%2F2%29+=+sin%28alpha%29%2F%281%2Bcos%28alpha%29%29+=+%281-cos%28alpha%29%29%2Fsin%28alpha%29,

sin%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, cos%28alpha%29+=+%281-tan%5E2%28alpha%2F2%29%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, tan%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281-tan%5E2%28alpha%2F2%29%29.

Three formulas in the last line above allow to represent/to express the major trigonometric functions sin%28alpha%29, cos%28alpha%29 and tan%28alpha%29 as the rational functions of tan%28alpha%2F2%29.
This implies that any rational function of sin%28alpha%29, cos%28alpha%29 and tan%28alpha%29 can be expressed as a rational function of tan%28alpha%2F2%29 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

cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2,

which is the same as the formula

cos%28alpha%2F2%29+=+sqrt%28%281%2Bcos%28alpha%29%29%2F2%29.

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:

cos%28gamma+%2B+delta%29+=+cos%28gamma%29%2Acos%28delta%29+-+sin%28gamma%29%2Asin%28delta%29.

Simply take gamma+=+alpha%2F2, delta+=+alpha%2F2 in this formula. You get

cos%28alpha%29+=+cos%5E2%28alpha%2F2%29+-+sin%5E2%28alpha%2F2%29.

Now, substitute here
sin%5E2%28alpha%2F2%29+=+1+-+cos%5E2%28alpha%2F2%29.

You get
cos%28alpha%29+=+cos%5E2%28alpha%2F2%29+-+%281+-+cos%5E2%28alpha%2F2%29%29%29+=+2%2Acos%5E2%28alpha%2F2%29+-+1.

The last step is to make a simple rearrangement in this formula
cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2,

which is exactly what we are going to prove.
The proof is completed.

The formula
cos%28alpha%2F2%29+=+sqrt%28%281%2Bcos%28alpha%29%29%2F2%29.

is the immediate consequence of the previous one.

Proof of the half argument formula for sines


We are going to prove the formula

sin%5E2%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2F2

which is the same as the formula

sin%28alpha%2F2%29+=+sqrt%28%281-cos%28alpha%29%29%2F2%29.

In the section above we just proved that

cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2.

Therefore,

sin%5E2%28alpha%2F2%29+=+1-cos%5E2%28alpha%2F2%29+=+1+-+%281%2Bcos%28alpha%29%29%2F2+=+%281-cos%28alpha%29%29%2F2,

which is exactly what we are going to prove.
The proof is completed.

The formula
sin%28alpha%2F2%29+=+sqrt%28%281-cos%28alpha%29%29%2F2%29.

is the immediate consequence of the previous one.

Proof of the half argument formula for tangents


We are going to prove the formula

tan%28alpha%2F2%29+=+sin%28alpha%29%2F%281%2Bcos%28alpha%29%29.

We have

tan%28alpha%2F2%29+=+sin%28alpha%2F2%29%2Fcos%28alpha%2F2%29+=+%282%2Asin%28alpha%2F2%29%2Acos%28alpha%2F2%29%29%2F%282%2Acos%28alpha%2F2%29%2Acos%28alpha%2F2%29%29     (after multiplying the numerator and denominator by 2cos%28alpha%2F2%29)

= sin%28alpha%29%2F%281%2Bcos%28alpha%29%29                                           (after applying the half argument formula for cosines cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2 proved above in this lesson).

This is exactly what we are going to prove.

Regarding the formula

tan%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2Fsin%28alpha%29,

the proof is very similar. We have

tan%28alpha%2F2%29+=+sin%28alpha%2F2%29%2Fcos%28alpha%2F2%29+=+%282%2Asin%28alpha%2F2%29%2Asin%28alpha%2F2%29%29%2F%282%2Acos%28alpha%2F2%29%2Asin%28alpha%2F2%29%29     (after multiplying the numerator and denominator by 2sin%28alpha%2F2%29)

= %281-cos%28alpha%29%29%2Fsin%28alpha%29                                           (after applying the half argument formula for sines sin%5E2%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2F2 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.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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
cos%28alpha+%2B+beta%29+=+cos%28alpha%29%2Acos%28beta%29+-+sin%28alpha%29%2Asin%28beta%29,
cos%28alpha+-+beta%29+=+cos%28alpha%29%2Acos%28beta%29+%2B+sin%28alpha%29%2Asin%28beta%29,
sin%28alpha+%2B+beta%29+=+sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29,
sin%28alpha+-+beta%29+=+sin%28alpha%29%2Acos%28beta%29+-+cos%28alpha%29%2Asin%28beta%29,

tan%28alpha+%2B+beta%29+=+%28tan%28alpha%29+%2B+tan%28beta%29%29%2F%281+-+tan%28alpha%29%2Atan%28beta%29%29, tan%28alpha+-+beta%29+=+%28tan%28alpha%29+-+tan%28beta%29%29%2F%281+%2B+tan%28alpha%29%2Atan%28beta%29%29.
    The lessons Addition and subtraction formulas and
                     Addition and subtraction formulas - Examples







Addition and subtraction of trigonometric functions
sin%28alpha%29+%2B+sin%28beta%29+=+2%2Asin%28%28alpha%2Bbeta%29%2F2%29%2Acos%28%28alpha-beta%29%2F2%29,

sin%28alpha%29+-+sin%28beta%29+=+2%2Asin%28%28alpha-beta%29%2F2%29%2Acos%28%28alpha%2Bbeta%29%2F2%29,

cos%28alpha%29+%2B+cos%28beta%29+=+2%2Acos%28%28alpha%2Bbeta%29%2F2%29%2Acos%28%28alpha-beta%29%2F2%29,

cos%28alpha%29+-+cos%28beta%29+=+-2%2Asin%28%28alpha%2Bbeta%29%2F2%29%2Asin%28%28alpha-beta%29%2F2%29,

tan%28alpha%29+%2B-+tan%28beta%29+=+sin%28alpha+%2B-+beta%29%2F%28cos%28alpha%29%2Acos%28beta%29%29, cot%28alpha%29+%2B-+cot%28beta%29+=+sin%28alpha+%2B-+beta%29%2F%28sin%28alpha%29%2Asin%28beta%29%29.
    The lessons Addition and subtraction of trigonometric functions and
                     Addition and subtraction of trigonometric functions - Examples












Product of trigonometric functions
sin%28alpha%29%2Asin%28beta%29+=+%281%2F2%29%2A%28cos%28alpha-beta%29+-+cos%28alpha%2Bbeta%29%29,

cos%28alpha%29%2Acos%28beta%29+=+%281%2F2%29%2A%28cos%28alpha-beta%29+%2B+cos%28alpha%2Bbeta%29%29,

sin%28alpha%29%2Acos%28beta%29+=+%281%2F2%29%2A%28sin%28alpha-beta%29+%2B+sin%28alpha%2Bbeta%29%29.
                                 The lessons Product of trigonometric functions and
                                                   Product of trigonometric functions - Examples






Powers of trigonometric functions
cos%5E2%28alpha%29+=+%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

sin%5E2%28alpha%29+=+-%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

cos%5E3%28alpha%29+=+%281%2F4%29%2Acos%283alpha%29+%2B+%283%2F4%29%2Acos%28alpha%29,

sin%5E3%28alpha%29+=+-%281%2F4%29%2Asin%283alpha%29+%2B+%283%2F4%29%2Asin%28alpha%29.
                                          The lessons Powers of Trigonometric functions and
                                                            Powers of Trigonometric functions - Examples









Trigonometric functions of multiply argument
cos%282alpha%29+=+2%2Acos%5E2%28alpha%29+-+1,

sin%282alpha%29+=+2%2Asin%28alpha%29%2Acos%28alpha%29,

cos%283alpha%29+=+4%2Acos%5E3%28alpha%29+-+3%2Acos%28alpha%29,

sin%283alpha%29+=+-4%2Asin%5E3%28alpha%29+%2B+3%2Asin%28alpha%29.
                                                The lessons Trigonometric functions of multiply argument and
                                                                Trigonometric functions of multiply argument - Examples








Trigonometric functions of half argument
sin%5E2%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2F2, cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2,

tan%28alpha%2F2%29+=+sin%28alpha%29%2F%281%2Bcos%28alpha%29%29+=+%281-cos%28alpha%29%29%2Fsin%28alpha%29,

sin%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, cos%28alpha%29+=+%281-tan%5E2%28alpha%2F2%29%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, tan%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281-tan%5E2%28alpha%2F2%29%29.
The lessons Trigonometric functions of half argument (this lesson) and
                  Trigonometric functions of half argument - Examples









Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems
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