Lesson Product of trigonometric functions

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Product of trigonometric functions


The formulas for the product of trigonometric functions are:

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.

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

Proof of the sines product formula


We are going to prove the formula

sin%28alpha%29%2Asin%28beta%29+=+%281%2F2%29%2A%28cos%28alpha-beta%29+-+cos%28alpha%2Bbeta%29%29.

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

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

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

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

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,

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 (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,

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 and
                  Trigonometric functions of half argument - Examples









Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems
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