Lesson Product of trigonometric functions - Examples

Algebra ->  Algebra  -> Trigonometry-basics -> Lesson Product of trigonometric functions - Examples      Log On

Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!
Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!

   

This Lesson (Product of trigonometric functions - Examples) was created by by ikleyn(4) About Me : View Source, Show
About ikleyn:

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.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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 (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
This lesson has been accessed 1011 times.