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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?


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$egingroup$ 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


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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.


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Why does simplifying $arcsin(x) -arcsin(y) = fracpi2$ to $y=-cos(arcsin(x))$ change the graph?
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