How to solve the equation arcsin xarccos1square root 2pi2Sol

How to solve the equation arcsin x+arccos(1/square root 2)=pi/2?

Solution

We\'ll keep the arcsin function to the left side and we\'ll shift arccos (1/sqrt2) to the right.

arcsin x = pi/2 - arccos (1/sqrt2)

We\'ll apply sine function both sides:

sin(arcsin x) = sin [pi/2 - arccos (1/sqrt2)] (1)

We\'ll use the formula:

sin(a-b) = sin a*cos b - sin b*cos a

According to the formula, we\'ll have:

sin [pi/2 - arccos (1/sqrt2)] = sin pi/2*cos(arccos (1/sqrt2)) - sin(arccos (1/sqrt2))*cos pi/2

By definition, the following terms are: sin(arcsin x) = x ; cos(arccos (1/sqrt2)) = 1/sqrt2 ; sin pi/2 = 1 and cos pi/2 = 0

We\'ll re-write (1):

x = 1*1/sqrt2 - 0

x = 1/sqrt 2

x = sqrt 2/2

The solution of the equation is x = sqrt 2/2.