Lesson Addition and subtraction of trigonometric functions

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Addition and subtraction of trigonometric functions


The addition and subtraction trigonometric functions formulas are:














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

Proof of the sines addition formula


We are going to prove the formula

.

The proof is very simple and straightforward.

Introduce two new angles by formulas
gamma+=+%28alpha+%2B+beta%29%2F2, delta+=+%28alpha+-+beta%29%2F2.

Note that
gamma+%2B+delta+=+%28alpha+%2B+beta%29%2F2+%2B+%28alpha+-+beta%29%2F2+=+alpha,

gamma+-+delta+=+%28alpha+%2B+beta%29%2F2+-+%28alpha+-+beta%29%2F2+=+beta.

Now, you can express sin%28alpha%29 as sin%28gamma+%2B+delta%29 using the addition formula for sines:
      (this addition formula was proved in the lesson Addition and subtraction formulas in this module)

and express sin%28beta%29 as sin%28gamma+-+delta%29 using the subtraction formula for sines:
      (this subtraction formula was proved in the lesson Addition and subtraction formulas in this module).

Add left sides and right sides of these two formulas. You get
sin%28alpha%29+%2B+sin%28beta%29+=+2%2Asin%28gamma%29%2Acos%28delta%29,

what is exactly what we were going to prove.

Proof of the sines subtraction formula


We are going to prove the formula

.

The proof is very similar to the previous one.

Use the same angles gamma and delta as before. Express sin%28alpha%29 as sin%28gamma+%2B+delta%29 using the same addition formula for sines:


and express sin%28beta%29 as sin%28gamma+-+delta%29 using the same subtraction formula for sines:
.

Now, subtract the second formula from the first one side by side.You get
sin%28alpha%29+-+sin%28beta%29+=+2%2Acos%28gamma%29%2Asin%28delta%29,

which is exactly what we were going to prove.

Proofs of the addition and subtraction formulas for cosines are very similar to that of sines.

Proof of the tangents addition formula


We are going to prove the formula



Start from the left side and transform it step by step:
  (after converting the fractions to the common denominator)

= sin%28alpha%2Bbeta%29%2F%28cos%28alpha%29%2Acos%28beta%29%29                                             (after using the formula of sines for the sum of angles)

This is exactly what we were going to prove.

Proof of the tangents subtraction formula


The proof is similar to the previous one.

For examples of applications of these formulas see the lesson Addition and subtraction 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,

, .
    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 (this lesson) and
                     Addition and subtraction of trigonometric functions - Examples












Product of trigonometric functions
,

,

.
                                 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









Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems
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