How do you find the exact value of #arcsin(sin(2))#?

Question

Violet Anderson · Accepted Answer

The exact value of #arcsin(sin(2))# is simply #2#.

Whenever we take the arcsin of sin, or the arccos of cos, or the inverse of any trig function, they always cancel each other out. So #arctan(tan(3))=3#, #arcsin(sin(1))=1#, and so on. The reason for this is because to find the arcsin of a given number, for instance, #arcsinx#, we are basically asking "When will the sin of some number equal #x#?" So with this problem, instead of #x# we have the arcsin of sin. Thus we are basically asking "When will the sin of some number equal #sin(2)#?" As an equation with our desired answer being n, that looks like #sin(n)=sin(2)# So #n# is clearly #2#. This also works no matter order we're taking #sin# and #arcsin# in. So for example, #sin(arcsin(0))=0#

Anna Andrews · Answer

undefined To find the exact value of arcsin(sin(2)), you need to consider the range of the arcsin function. Since arcsin returns angles between -π/2 and π/2 (or -90 degrees and 90 degrees), you need to find the angle within this range whose sine is 2. However, since the sine function's range is between -1 and 1, it's impossible for sin(2) to equal 2. Therefore, there is no real solution for arcsin(sin(2)).