Lesson Powers of trigonometric functions

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Powers of trigonometric functions


The formulas for Powers of trigonometric functions are:

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.

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

Proof of the cosines square formula


We are going to prove the formula

cos%5E2%28alpha%29+=+%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2.

The proof is very simple and straightforward.

It is based on the addition formula for cosines of the lesson Addition and subtraction formulas in this module:

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

Simply take beta+=+alpha in this formula. You get

cos%282alpha%29+=+cos%5E2%28alpha%29+-+sin%5E2%28alpha%29.

Now, substitute
sin%5E2%28alpha%29+=+1+-+cos%5E2%28alpha%29.

to the previous equation. You get
.

Make simple rearrangements in the line above, and you get exactly what we are going to prove.
The proof is completed.

Proof of the sines square formula


We are going to prove the formula

sin%5E2%28alpha%29+=+-%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2.

Start from
sin%5E2%28alpha%29+=+1+-+cos%5E2%28alpha%29,

which is kind of the basic formulas.
Substitute
cos%5E2%28alpha%29+=+%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

the formula which was proved above. You get
.

Make simple rearrangements in the line above, and you get exactly what we are going to prove.
The proof is completed.

Proof of the cosines cube formula


We are going to prove the formula

cos%5E3%28alpha%29+=+%281%2F4%29%2Acos%283alpha%29+%2B+%283%2F4%29%2Acos%28alpha%29.

Let us apply the addition formula for cosines of the lesson Addition and subtraction formulas of this module
in the form

. (*)

For cos%282alpha%29 we just have ready the expressions
cos%282alpha%29+=+2%2Acos%5E2%28alpha%29+-+1,
which was proved above.

For sin%282alpha%29 the general addition formula for sines gives
.

Now, substitute these expressions for cos%282alpha%29 and sin%282alpha%29 to the formula (*) above. You get

            =
            =.

Make the last rearrangements in the lines above, and you get exactly what we are going to prove.
The proof is completed.

For examples of applications of these formulas see the lesson Powers 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 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 (this lesson) 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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