Attention required!


$sin^3x+cos^3x=1\(sin x+cos x)(sin^2x-sin xcdotcos x+cos^2x)=1\(sin x+cos x)(1-sin xcdotcos x)=1$

What should I bởi next?




Both of $sin(x),cos(x)$ must be nonnegative since, if one of them was negative, the equation $sin^3(x)+cos^3(x)=1$ would imply that the other one is more than $1$, contradiction.Thus, we have $0le sin(x) le 1$ and $0le cos(x)le 1$.If both of $sin(x),cos(x)$ are less than $1$, then, since they are both nonnegative, we would have $$egincases0le sin^3(x) but that would imply$$sin^3(x)+cos^3(x) contradiction.It follows that one of $sin(x),cos(x)$ must be equal to $1$, & the other must be equal khổng lồ zero.From that information, I"m sure you can finish the solution.

Bạn đang xem: Attention required!



Let $sin x+cos x=timplies t^2=?$

$$1=dfract2-(t^2-1)2iff t^3-3t+2=0$$

Clearly, $t=1$ a solution


Let use by $t= an (x/2)$

$sin x=frac2t1+t^2$

$cos x=frac1-t^21+t^2$

to obtain


$$iff t^2(t^4-4t+3)=t^2(t-1)^2(t^2+2t+3)=0$$

and since $t^2+2t+3>0$ the solutions are

$t=0 implies frac x 2=kpiimplies x=2kpi$

$t=1 implies frac x 2=fracpi4+kpi implies x=fracpi2+2kpi $

As an alternative by

$$sin^3 x+ cos ^3 x=1 iff sin xcdot sin^2+cos xcdot cos^2 x=1$$

since $sin^2 x+ cos ^2 x=1$, the given equality is a weighted mean of $sin x$ and $cos x$ which holds if & only if

$sin x=1,,cos x=0$


$cos x=1,,sin x=0$

Thanks for contributing an answer lớn cameraminhtan.vnematics Stack Exchange!

Please be sure khổng lồ answer the question. Provide details & share your research!

But avoid

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

Xem thêm: Người Về Tìm Chi Khi Màu Mắt Đã Tàn Úa, Lời Bài Hát Vòng Tay Lỡ Làng

Use cameraminhtan.vnJax to format equations. cameraminhtan.vnJax reference.

Xem thêm: Đuổi Việc Cô Giáo Nguyễn Thị Thu Trang Mới Nhất, Tin Tức Cô Giáo Nguyễn Thị Thu Trang Mới Nhất

To learn more, see our tips on writing great answers.

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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

Site kiến thiết / logo sản phẩm © 2022 Stack Exchange Inc; user contributions licensed under cc by-sa. Rev2022.6.30.42496

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device và disclose information in accordance with our Cookie Policy.