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


Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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