Lesson Addition and subtraction formulas

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Addition and subtraction formulas


The addition and subtraction Trigonometry formulas are:

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





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

Proof of the addition formula for cosines


In the unit circle consider the point P1 with the central angle -alpha  
(coordinates (x%5B1%5D, y%5B1%5D), see the Figure 1a).
Consider also the point P2 with the central angle beta (coordinates
(x%5B2%5D, y%5B2%5D), see the Figure 1a).
Let P3 be the point with the central angle alpha%2Bbeta (coordinates
(x%5B3%5D, y%5B3%5D), see the Figure 1b).
We have
x%5B1%5D+=+cos%28alpha%29, y%5B1%5D+=+sin%28-alpha%29=-sin%28alpha%29,   (1)
x%5B2%5D+=+cos%28beta%29, y%5B2%5D+=+sin%28beta%29,                   (2)
x%5B3%5D+=+cos%28alpha%2Bbeta%29, y%5B3%5D+=+sin%28alpha%2Bbeta%29.         (3)

Figure 1a. Proof of the addition formula
             for cosines


Figure 1b. Proof of the addition formula
             for cosines
Since triangles P1OP2 and AOP3 are congruent, the segment [P1,P2] (Figure 1a) has the same length as the segment [A,P3] (Figure 1b),
where A is the point with coordinates (1,0). This gives you the equation
.
Simplify this equation step by step. You get
(after opening the brackets),
-2x%5B1%5Dx%5B2%5D+-+2x%5B1%5Dx%5B2%5D+=+-2x%5B3%5D                                                     (after using x%5B1%5D%5E2%2By%5B1%5D%5E2=1, x%5B2%5D%5E2%2By%5B2%5D%5E2=1 and x%5B3%5D%5E2%2By%5B3%5D%5E2=1),
x%5B1%5Dx%5B2%5D+%2B+y%5B1%5Dy%5B2%5D+=+x%5B3%5D                                                             (after dividing both sides by -2).

Substituting expressions (1), (2) and (3) for x%5B1%5D, y%5B1%5D, x%5B2%5D, y%5B2%5D, x%5B3%5D and y%5B3%5D, you get exactly the addition formula
cos%28alpha+%2B+beta%29+=+cos%28alpha%29%2Acos%28beta%29+-+sin%28alpha%29%2Asin%28beta%29.

The proof is completed.

Proof of the subtraction formula for cosines


Now, when the addition formula for cosines is proved, the proof of the subtraction formula can be made in couple of lines. Simply introduce the angle gamma+=+-beta and then apply the addition formula for cosines. Use cos%28gamma%29+=+cos%28beta%29, sin%28gamma%29+=+-sin%28beta%29:
=
.

The proof is completed.

Proof of the addition formula for sines


You can easy get the addition formula for sines from the subtraction formula for cosines, which is already proved. Simply use the reduction formulas
sin%28alpha%29+=+cos%28pi%2F2+-+alpha%29, cos%28alpha%29+=+sin%28pi%2F2+-+alpha%29
(see, for example, the lesson The Amazing Unit Circle: Trigonometric Identities of this module).

You have
=
cos%28pi%2F2+-+alpha%29%2Acos%28beta%29+%2B+sin%28pi%2F2+-+alpha%29%2Asin%28beta%29 =
sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29.

The proof is completed.

Proof of the subtraction formula for sines


Similarly, you can easy get the subtraction formula for sines from the addition formula for cosines, which is already proved.
Simply use the same reduction formula as in the previous proof.

=
cos%28pi%2F2+-+alpha%29%2Acos%28beta%29+-+sin%28pi%2F2+-+alpha%29%2Asin%28beta%29 =
sin%28alpha%29%2Acos%28beta%29+-+cos%28alpha%29%2Asin%28beta%29.

The proof is completed.

Proof of the addition and subtraction formulas for tangents


Now, when the addition and subtraction formulas for cosines and sines are proved, the proof of the addition and subtraction formulas
for tangents is straightforward.

For addition you have

tan%28alpha+%2B+beta%29+=+sin%28alpha+%2B+beta%29%2Fcos%28alpha+%2B+beta%29 =

=     (after dividing both numerator and denominator by cos%28alpha%29%2Acos%28beta%29)

= %28tan%28alpha%29+%2B+tan%28beta%29%29%2F%281+-+tan%28alpha%29%2Atan%28beta%29%29.

The proof is completed.

For subtraction you have

tan%28alpha+-+beta%29+=+sin%28alpha+-+beta%29%2Fcos%28alpha+-+beta%29 =

=     (after dividing both numerator and denominator by cos%28alpha%29%2Acos%28beta%29)

= %28tan%28alpha%29+-+tan%28beta%29%29%2F%281+%2B+tan%28alpha%29%2Atan%28beta%29%29.

The proof is completed.

For examples see the lesson Addition and subtraction formulas - 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 (this lesson) 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






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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