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What is the property that tells us \$arcsin(-x) = -arcsin(x)\$ ?

I"ve also seen in an exercise that : \$arcsin (sin(2x)) = 2x ~~\$ if \$x inleft<0,fracpi4 ight>\$

And what about: \$arcsin(sin(2x)) = pi - 2x ~~\$ if \$x in left>fracpi4,fracpi2 ight>\$?

What justifies these relations?  \$egingroup\$ None of the 17 questions you have asked has an accepted answer. With all due respect, vì you think any one of them helpful khổng lồ you? \$endgroup\$
\$sin(arcsin(-x))=-x, sin(-arcsin(x))=-sin(arcsin(x))=-x\$, since \$sin:<-pi/2,pi/2> ightarrow <-1,1>\$ is the inverse of \$arcsin\$ we deduce that \$arcsin(-x)=-arcsin(x)\$ since the restriction of \$sin\$ on \$<-pi/2,pi/2>\$ is injective.

If \$xin , 2xin \$ and \$pi-2xin <0,pi/2>\$, since\$sin(pi-2x)=sin(2x)\$ và \$sin\$ is the inverse of \$arcsin:<-1,1> ightarrow <-pi/2,pi/2>\$, we deduce that \$pi-2x\$ is the unique element of \$<-pi/2,pi/2>\$ such that \$sin(pi-2x)=sin(2x)\$ & \$arcsin(2x)=pi-2x\$.

Xem thêm: Giải Toán 7 Bài Tập Cộng Trừ Đa Thức Và Bài Tập, Giải Toán 7 Bài 6: Cộng, Trừ Đa Thức We can also write:\$\$sin(x) = frace^ix-e^-ix2i\$\$So, if you"re more familiar with the exponential function, it can b easier to lớn derive identities with this. ## Not the answer you're looking for? Browse other questions tagged trigonometry or ask your own question.

Why does simplifying \$arcsin(x) -arcsin(y) = fracpi2\$ to \$y=-cos(arcsin(x))\$ change the graph? Site kiến thiết / hình ảnh © 2022 Stack Exchange Inc; user contributions licensed under cc by-sa. Rev2022.4.21.42004