Lesson Trigonometric functions of half argument - Examples

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Trigonometric functions of half argument - Examples


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.

The proofs of these formulas are presented in the lesson  Trigonometric functions of half argument  in this module.
Below are examples of applications of these formulas.

Problem 1

Find  sin(22°30'),  sin(22°30'),  tan(22°30').

Solution

First,  calculate  sin(22°30').
Since  22°30' = 45°/2=pi%2F8,  you can apply the formula of half argument for sines  (see the lesson  Trigonometric functions of half argument  in this module):
sin(22°30') = sqrt%28%281-cos%28pi%2F4%29%29%2F2%29 = sqrt%28%281-sqrt%282%29%2F2%29%2F2%29 = sqrt%28%282-sqrt%282%29%29%2F4%29 = sqrt%282-sqrt%282%29%29%2F2.

Similarly,  cos(22°30') = sqrt%28%281%2Bcos%28pi%2F4%29%29%2F2%29 = sqrt%28%281%2Bsqrt%282%29%2F2%29%2F2%29 = sqrt%28%282%2Bsqrt%282%29%29%2F4%29 = sqrt%282%2Bsqrt%282%29%29%2F2.

Hence,
tan(22°30') = sin(22°30')/cos(22°30') = sqrt%28%282-sqrt%282%29%29%2F%282%2Bsqrt%282%29%29%29 = (further transformations) = = sqrt%28+%28%282-sqrt%282%29%29%5E2%29%2F%282%5E2-%28sqrt%282%29%29%5E2%29+%29 = %282-sqrt%282%29%29%2Fsqrt%282%29 = sqrt%282%29-1.

Problem 2

Find  cos(15°),  sin(15°),  tan(15°).

Solution

First,  find sin(15°).  Use the formula of half argument for sines (see the lesson  Trigonometric functions of half argument  in this module):
sin^2(15°) = (1-cos(30°)/2 = %281-sqrt%283%29%2F2%29%2F2 = %282-sqrt%283%29%29%2F4.

Hence,
sin(15°) = sqrt%282-sqrt%283%29%29%2F2.

Now,  find cos(15°).  Use formula for cosines of half argument,  which is the second formula in this lesson.  You have

cos^2(15°) = (1+cos(30°)/2 = %281%2Bsqrt%283%29%2F2%29%2F2 = %282%2Bsqrt%283%29%29%2F4.

Hence,
cos(15°) = sqrt%282%2Bsqrt%283%29%29%2F2.

Consequently,
tan(15°) = sin(15°)/cos(15°) = sqrt%282-sqrt%283%29%29%2Fsqrt%282%2Bsqrt%283%29%29 = (further transformations) = = sqrt%28+%28%282-sqrt%283%29%29%5E2%29%2F%282%5E2+-+%28sqrt%283%29%29%5E2%29+%29 = +%282-sqrt%283%29%29%2F+sqrt%284-3%29 = %282-sqrt%283%29%29%2F+sqrt%281%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,

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    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






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 (this lesson)









Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems


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