Lesson Addition and subtraction of trigonometric functions - Examples

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Addition and subtraction of trigonometric functions - Examples


The addition and subtraction trigonometric functions formulas are:

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 proofs of these formulas are presented in the lesson Addition and subtraction of trigonometric functions in this module.
Below are examples of application of these formulas.

Example 1
Find sin(75°) + sin(15°).

Solution
Use the addition formula for sines:
sin%28alpha%29+%2B+sin%28beta%29+=+2%2Asin%28%28alpha%2Bbeta%29%2F2%29%2Acos%28%28alpha-beta%29%2F2%29.

You have
sin(75°) + sin(15°) = 2*sin((75°+15°)/2)*cos((75°-15°)/2) = 2*sin(45°)*cos(30°) = 2%2Asqrt%282%29%2F2%2Asqrt%283%29%2F2+=+sqrt%286%29%2F2.

Example 2
Prove that %28sin%28alpha%29%2Bsin%283alpha%29%29%2F%28cos%28alpha%29%2Bcos%283alpha%29%29+=+tan%282alpha%29.

Solution
Use the addition formula for sines and cosines:
sin%28alpha%29+%2B+sin%28beta%29+=+2%2Asin%28%28alpha%2Bbeta%29%2F2%29%2Acos%28%28alpha-beta%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.

You have
%28sin%28alpha%29%2Bsin%283alpha%29%29%2F%28cos%28alpha%29%2Bcos%283alpha%29%29+=+2%2Asin%28%28alpha%2B3alpha%29%2F2%29%2Acos%28%28alpha-3alpha%29%2F2%29%2F%282%2Acos%28%28alpha%2B3alpha%29%2F2%29%2Acos%28%28alpha-3alpha%29%2F2%29%29%29 = 2%2Asin%282alpha%29%2Acos%28-2alpha%29%2F%282%2Acos%282alpha%29%2Acos%28-2alpha%29%29+=+sin%282alpha%29%2Fcos%282alpha%29+=+tan%282alpha%29.

The proof is completed.

Example 3
Prove that 1+%2B+cos%282alpha%29+%2B+cos%284alpha%29+%2B+cos%286alpha%29+=+4%2Acos%28alpha%29%2Acos%282alpha%29%2Acos%283alpha%29.

Solution
Using the addition formula for cosines you have
1+%2B+cos%282alpha%29+=+cos%280%29+%2B+cos%282alpha%29+=+2%2Acos%28%280%2B2alpha%29%2F2%29%2Acos%28%280-2alpha%29%2F2%29+=+2%2Acos%5E2%28alpha%29,

cos%284alpha%29+%2B+cos%286alpha%29+=+2%2Acos%28%284alpha%2B6alpha%29%2F2%29%2Acos%28%284alpha-6alpha%29%2F2%29+=+2%2Acos%285alpha%29%2Acos%28-alpha%29+=+2%2Acos%28alpha%29%2Acos%285alpha%29.

By summing the left and the right sides of these two equalities, you get
1+%2B+cos%282alpha%29+%2B+cos%284alpha%29+%2B+cos%286alpha%29+=+2%2Acos%5E2%28alpha%29+%2B+2%2Acos%28alpha%29%2Acos%285alpha%29.

Furthermore, you can transform the right side as follows:
2%2Acos%5E2%28alpha%29+%2B+2%2Acos%28alpha%29%2Acos%285alpha%29+=+2%2Acos%28alpha%29%2A%28cos%28alpha%29+%2B+cos%285alpha%29%29

= 2%2Acos%28alpha%29%2A%28cos%28alpha%2B5alpha%29%2F2%29%2A%28cos%28alpha-5alpha%29%2F2%29 = 2%2Acos%28alpha%29%2Acos%283alpha%29%2Acos%28-2alpha%29+=+2%2Acos%28alpha%29%2Acos%282alpha%29%2Acos%283alpha%29.

The proof is completed.

Example 4
Prove yourself that 1+-+cos%282alpha%29+%2B+cos%284alpha%29+-+cos%286alpha%29+=+4%2Asin%28alpha%29%2Acos%282alpha%29%2Asin%283alpha%29.

Solution
The proof is similar to that of the Example 3.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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 (this lesson)












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 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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