Math, asked by MohamedSuhaib, 1 year ago

expand sin^6 theta interms of cos theta​

Answers

Answered by Anonymous
0

Answer:

sin

6

(

x

)

=

1

32

(

cos

(

6

x

)

6

cos

(

4

x

)

+

15

cos

(

2

x

)

10

)

Step-by-step explanation:

We want to expand

sin

6

(

x

)

One way is to use these identities repeatedly

sin

2

(

x

)

=

1

2

(

1

cos

(

2

x

)

)

cos

2

(

x

)

=

1

2

(

1

+

cos

(

2

x

)

)

This often gets quite long, which sometimes leads to mistake

Another way is to use the complex numbers (and Euler's formula)

We can express sine and cosine as

sin

(

x

)

=

e

i

x

e

i

x

2

i

and

cos

(

x

)

=

e

i

x

+

e

i

x

2

Thus

sin

6

(

x

)

=

(

e

i

x

e

i

x

2

i

)

6

=

(

e

i

x

e

i

x

)

6

64

=

e

6

i

x

6

e

4

i

x

+

15

e

2

i

x

20

+

15

e

2

i

x

6

e

4

i

x

+

e

6

i

x

64

=

1

32

e

6

i

x

+

e

6

i

x

6

e

4

i

x

6

e

4

i

x

+

15

e

2

i

x

+

15

e

2

i

x

20

2

=

1

32

(

cos

(

6

x

)

6

cos

(

4

x

)

+

15

cos

(

2

x

)

10

)

The third way is using De Moivre's theorem, we can express

2

cos

(

n

x

)

=

z

n

+

1

z

n

and

2

i

sin

(

n

x

)

=

z

n

1

z

n

where

z

n

=

(

cos

(

x

)

+

i

sin

(

x

)

)

n

=

cos

(

n

x

)

+

i

sin

(

n

x

)

Thus

(

2

i

sin

(

x

)

)

6

=

(

z

1

z

)

6

sin

6

(

x

)

=

1

64

(

z

1

z

)

6

Expand the binomial on the right hand side

B

I

N

=

(

z

6

+

1

z

6

6

z

4

6

z

4

+

15

z

2

+

15

z

2

20

)

R

H

S

=

(

z

6

+

1

z

6

6

(

z

4

+

1

z

4

)

+

15

(

z

+

1

z

)

+

20

)

R

H

S

=

(

2

cos

(

6

x

)

12

cos

(

4

x

)

+

30

cos

(

2

x

)

20

)

Thus

sin

6

(

x

)

=

1

64

(

2

cos

(

6

x

)

12

cos

(

4

x

)

+

30

cos

(

2

x

)

20

)

sin

6

(

x

)

=

1

32

(

cos

(

6

x

)

6

cos

(

4

x

)

+

15

cos

(

2

x

)

10

)

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