Lesson Product of trigonometric functions

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Product of trigonometric functions


The formulas for the product of trigonometric functions are:

,

,

.

In this lesson you can learn how to prove these formulas.

Proof of the sines product formula


We are going to prove the formula

.

The proof is very simple and straightforward.

It is based on the addition and subtraction formulas for cosines from the lesson  Addition and subtraction formulas  in this module:

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.

Subtract the first formula from the second one side by side.  You get

2%2Asin%28alpha%29%2Asin%28beta%29+=+cos%28alpha+-+beta%29+-+cos%28alpha+%2B+beta%29.

Now,  divide both sides of the last equality by  2.
You got what we were going to prove.  The proof is completed.

Proof of the cosines product formula


We are going to prove the formula

.

Use the same addition and subtraction formulas for cosines of the lesson  Addition and subtraction formulas  in this module:

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.

Now,  add the first formula to the second one side by side.  You get

2%2Acos%28alpha%29%2Acos%28beta%29+=+cos%28alpha+%2B+beta%29+%2B+cos%28alpha+-+beta%29.

Divide both sides of the last equality by  2.
You got what we were going to prove.  The proof is completed.

Proof of the sines-cosines product formula


We are going to prove the formula

,

Now,  use the addition and subtraction formulas for sines of the lesson  Addition and subtraction formulas  of this module:

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.

Add the first formula to the second one side by side. You get

2%2Asin%28alpha%29%2Acos%28beta%29+=+sin%28alpha+%2B+beta%29+%2B+sin%28alpha+-+beta%29.

Divide both sides of the last equality by  2.
You got what we were going to prove.  The proof is completed.

For examples of applications of these formulas see the lesson  Product of trigonometric functions - 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  (this lesson)  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,

,

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  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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