Solve for x?

#arcsin(sqrt2x)=arccos(sqrtx)#

1 Answer
Mar 19, 2018

#x = 1/2#

Explanation:

Given #arcsin(sqrt2x)=arccos(sqrtx)#

Please observe that #x>=0# is implied by #sqrtx#

#arcsin(sqrt2x)=arccos(sqrtx), x >=0#

Eliminate the inverse sine function on the left by taking the sine of both sides:

#sqrt2x = sin(arccos(sqrtx)), x>=0#

Using the unit circle identity #sin(arccos(u)) = sqrt(1-u^2)# where

#sqrt2x = sqrt(1-u^2)#

where #u = sqrtx#:

#sqrt2x = sqrt(1-(sqrtx)^2), x>=0#

Simplify the right side:

#sqrt2x = sqrt(1-x), x>=0#

square both sides:

#2x^2 = 1-x, x >=0#

#2x^2+x -1=0, x>=0#

Factor:

#(2x-1)(x+1)=0, x>=0#

Because of #x >= 0, we discard the negative root:

#x = 1/2#