Lesson Product of trigonometric functions - Examples

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Product of trigonometric functions - Examples


The formulas for the product of trigonometric functions formulas are:

,

,

.

The proofs of these formulas are presented in the lesson  Product of trigonometric functions  in this module.
Below are examples of application of these formulas.

Example

Find  sin(15°),  cos(15°),  tan(15°).

Solution

First,  calculate  sin(15°).  Use the product formula for sines-sines:
.

Take  alpha = 15°.  You have
sin%5E2%28alpha%29+=+%281%2F2%29%2A%28cos%28alpha-alpha%29+-+cos%28alpha%2Balpha%29%29 = (1/2)*(cos(0°) - cos(30°)) = %281%2F2%29%2A%281-sqrt%283%29%2F2%29+=+%282-sqrt%283%29%29%2F4,
hence
sin(15°) = sqrt%282-sqrt%283%29%29%2F2.

Compare this with the formula
sin(15°) = %28sqrt%286%29-sqrt%282%29%29%2F4,
which was obtained in the lesson  Addition and subtraction formulas - Examples  in this module.

Let me show you that the number  %28sqrt%286%29-sqrt%282%29%29%2F4  is equal to  sqrt%282-sqrt%283%29%29%2F2.

Indeed,  the square of  %28sqrt%286%29-sqrt%282%29%29%2F4  is equal to  %286+-+2sqrt%2812%29+%2B+2%29%2F16+=+%288-4sqrt%283%29%29%2F16+=+%282-sqrt%283%29%29%2F4,  exactly as the square of  sqrt%282-sqrt%283%29%29%2F2,  so the original numbers are the same.


Now,  calculate cos(15°).  Use the product formula for cosines-cosines:
.

Take  alpha = 15°.  You have
cos%5E2%28alpha%29+=+%281%2F2%29%2A%28cos%28alpha-alpha%29+%2B+cos%28alpha%2Balpha%29%29 = (1/2)*(cos(0°) + cos(30°)) = %281%2F2%29%2A%281%2Bsqrt%283%29%2F2%29+=+%282%2Bsqrt%283%29%29%2F4,
hence
cos(15°) = sqrt%282%2Bsqrt%283%29%29%2F2.

Compare this with the formula
cos(15°) = %28sqrt%286%29%2Bsqrt%282%29%29%2F4,
which was obtained in the lesson  Addition and subtraction formulas - Examples  in this module.

Let me show you that the number  %28sqrt%286%29%2Bsqrt%282%29%29%2F4  is equal to  sqrt%282%2Bsqrt%283%29%29%2F2.

Indeed,  the square of  %28sqrt%286%29%2Bsqrt%282%29%29%2F4  is equal to  ,  exactly as the square of  sqrt%282%2Bsqrt%283%29%29%2F2,  so the original numbers are the same.


Now,  having ready expressions for  sin(15°)  and  cos(15°),  you can easily calculate  tan(15°)  as the fraction  sin(15°)/cos(15°):
tan(15°) = %28sqrt%286%29-sqrt%282%29%29%2F%28sqrt%286%29%2Bsqrt%282%29%29.

Note that,  as it was shown in the lesson  Addition and subtraction formulas - Examples  in this module,
%28sqrt%286%29-sqrt%282%29%29%2F%28sqrt%286%29%2Bsqrt%282%29%29+=+2-sqrt%283%29.



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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
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
,

,

,

,

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












Product of trigonometric functions
,

,

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                                 The lessons  Product of trigonometric functions  and
                                                     Product of trigonometric functions - Examples  (this lesson)






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


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