\$sin^4x+cos^4x\$I should rewrite this expression into a new size to plot the function.

Bạn đang xem: Trigonometry

eginalign& = (sin^2x)(sin^2x) - (cos^2x)(cos^2x) \& = (sin^2x)^2 - (cos^2x)^2 \& = (sin^2x - cos^2x)(sin^2x + cos^2x) \& = (sin^2x - cos^2x)(1) longrightarrow,= sin^2x - cos^2xendalign

Is that true?

eginalignsin^4 x +cos^4 x&=sin^4 x +2sin^2xcos^2 x+cos^4 x - 2sin^2xcos^2 x\&=(sin^2x+cos^2 x)^2-2sin^2xcos^2 x\&=1^2-frac12(2sin xcos x)^2\&=1-frac12sin^2 (2x)\&=1-frac12left(frac1-cos 4x2 ight)\&=frac34+frac14cos 4xendalign

Let \$\$displaystyle y=sin^4 x+cos^4 x = left(sin^2 x+cos^2 x ight)^2-2sin^2 xcdot cos^2 x = 1-frac12left(2sin xcdot cos x ight)^2\$\$

Now using \$\$ sin 2A = 2sin Acos A\$\$

So, we get \$\$displaystyle y=1-frac12sin^2 2x\$\$

Note that \$a^2 + b^2 = (a+b)^2 - 2ab\$

\$\$(sin^2 x)^2 + (cos^2 x)^2 = (sin^2 x + cos^2 x)^2 - 2sin^2 xcos^2 x =(sin^2 x + cos^2 x)^2 - 2(sin xcos x)^2 = \ 1 -frac sin^2 2x2\$\$

Note the following results:

\$\$ sin^2 x + cos^2 x = 1\$\$

\$\$ sin x cos x = fracsin 2x2\$\$

Expand in terms of complex exponentials.

\$\$sin^4 x + cos^4 x = left( frace^ix - e^-ix2i ight)^4 + left( frace^ix + e^-ix2 ight)^4\$\$

Notice that \$i^4 = +1\$, so we get

\$\$sin^4 x + cos^4 x = frac116 left( 2e^4ix + 2 e^-4ix + 12 ight)\$\$

where we use the relation \$(a+b)^4 = a^4 + 4 a^3 b + 6 a^2 b^2 + 4 ab^3 + b^4\$. The terms of the size \$a^3 b\$ và \$ab^3\$ all cancel by addition.

This leaves us with a final result:

\$\$sin^4 x + cos^4 x = frac416 left(frace^4ix + e^-4ix2 ight) + frac1216 = frac34 + frac14 cos 4x\$\$

share
Cite
Follow
answered Sep 30, 2015 at 17:14
MuphridMuphrid
\$endgroup\$
1
\$egingroup\$
If you want to lớn express in functions of higher frequencies like this \$\$sum_k=0^N sin(kx) + cos(kx)\$\$ Then you can use the Fourier transform together with convolution theorem. This will work out for any sum of powers of cos & sin, even \$sin^666(x)\$.

mô tả
Cite
Follow
answered Sep 30, năm ngoái at 17:09
\$endgroup\$
địa chỉ a comment |
0
\$egingroup\$
egineqnarraysin^4x + cos^ 4x &=& sin 4x + cos 4x+2 cos 2x sin 2x-2 cos 2x sin 2x \ &=& ( sin 2x + cos 2x)^2 -2 cos 2 sin 2 \ &=& 1-2 cos 2x sin 2x \&& (1-racine de 2 foi cos x sin x)(1+racine de 2 foi cos x sin x)endeqnarray

nội dung
Cite
Follow
edited Apr 16, 2017 at 20:14
Nosrati
answered Apr 16, 2017 at 19:14
\$endgroup\$
showroom a phản hồi |

Thanks for contributing an answer to orsini-gotha.comematics Stack Exchange!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

Use orsini-gotha.comJax to format equations. orsini-gotha.comJax reference.

Draft saved

Submit

### Post as a guest

Name
thư điện tử Required, but never shown

### Post as a guest

Name
thư điện tử

Required, but never shown

## Not the answer you're looking for? Browse other questions tagged trigonometry or ask your own question.

8
Deriving an expression for \$cos^4 x + sin^4 x\$
0
Find \$int_0^2pi frac1sin^4x + cos^4x dx\$.

Xem thêm: Soạn Bài Quá Trình Tạo Lập Văn Bản Ngữ Văn Lớp 7 Ngắn Gọn, Soạn Bài Quá Trình Tạo Lập Văn Bản (Trang 45)

Related
1
Trigonometric Identities: \$fracsin^2 heta1+cos heta=1-cos heta\$
2
Simplifying second derivative using trigonometric identities
1
Simplify \$-2sin(x)cos(x)-2cos(x)\$
0
Simplify the expression & leave answer in terms of \$sin x\$ and/or \$cos x\$
0
How can we bound \$fracsin( heta)cos( heta)\$
1
Minimum value of \$cos^2 heta-6sin heta cos heta+3sin^2 heta+2\$
0
Transforming the equation \$cot x -cos x = 0\$ into the size \$cos x(1- sin x) = 0\$
1
Simplify: \$fracsin(3x-y)-sin(3y-x)cos(2x)+cos(2y) \$
3
Simplify trigonometric expression using trigonometric identities
Hot Network Questions more hot questions

Question feed