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

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

and

as the rational functions of tan%28alpha%2F2%29

.
This implies that any rational function of sin%28alpha%29

and

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

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

.

Simply take gamma+=+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

.

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,

,

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

(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

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

,

the proof is very similar. We have

(after multiplying the numerator and denominator by 2sin%28alpha%2F2%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

The lessons Addition and subtraction formulas and
Addition and subtraction formulas - Examples

Addition and subtraction of trigonometric functions

The lessons Addition and subtraction of trigonometric functions and
Addition and subtraction of trigonometric functions - Examples

Product of trigonometric functions

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

The lessons Trigonometric functions of half argument (this lesson) and
Trigonometric functions of half argument - Examples

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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