Question

Sin inverse x + sin inverse 2x = 2π/3 solve for X

Answer 1

Answer:

let  

m = arcsin(x)  

thus:  

arccos(x) = π/2 - m  

similarly,  

if  

n = arcsin(y)  

then:  

arccos(y) = π/2 - n  

we have the two equation as  

m + n = 2π/3  

(π/2 - m) + (π/2 - n) = π/3  

the second equation simplifies to:  

m + n = 2π/3  

(the same equation)  

there are infinite solutions such that m = - n + 2π/3.  

that is:  

arcsin(x) = 2π/3 - arcsin(y)  

taking sin of both sides:  

x = sin(2π/3 - arcsin(y))  

recall:  

sin(A - B) = sinAcosB - sinBcosA  

we have:  

x = sin(2π/3)cos(arcsin(y)) - sin(arcsin(y))cos(2π/3)  

which simplifies to:  

x = (√3/2)cos(arcsin(y)) - y(-1/2)  

or:  

x = (√3/2)cos(arcsin(y)) + (1/2)y  

let  

arcsin(y) = m  

then  

y = sin(m) = y/1 = opposite/hypotenuse,  

and with the pythagorean theorem derive:  

adjacent = ± √(1 - y²)  

thus,  

cos(m) = ± √(1 - y²)  

and finally we have:  

x = ± (√3/2)√(1 - y²) + (1/2)y

Step-by-step explanation: