arcsin(x) means "The angle whose sine is x between and
arccos(x) means "The angle whose cosine is x" between and
"CoSine" means "Complement's Sine
We must consider two cases.
Case 1: when x is positive
Then the arcsin(x) and arcos(x) are both in QI
So x will be
90°-the angle whose sine is x.
But since we are using radians instead of degrees, it is:
- the angle whose sine is x.
So
and therefore
becomes
Multiply through by LCD of 2 to clear the fraction:
Therefore the Case 1 solution is x=1 since
Case 2: when x is negative
Then the arcsin(x) is a negative angle in QIV
And so 2arcsin(x) is an even more negative angle than arcsin(x)
arccos(x) is a positive angle in QII less than
So 2arcsin(x)+cos(x) is the sum of a positive angle less than
pi, and a negative angle. The sum of a positive angle less
than added to a negative angle can never equal to ,
so there is no solution to case 2.
Thus the only solution is x=1.
Edwin
