Pre calc question

Question

Write as an algebraic expression (no trigonometric functions) Assume x and y are positive and in the domain of the given inverse function cos(sin^-1 𝑥 − cos^-1 𝑦)

Answer 1

ONE

cos(sin^-1(x) - cos^-1(y)) = cos(theta-theta) - cos(0) = 1 or

algebraically, y=1

But you may have intended to write cos(sin^-1(y) - cos^-1(x))

but interpret either way as the angle whose sine is the opposite side over the hypotenuse

and the angle whose cosine is the adjacent side over the hypotenuse, to get the same answer = 1

r^2 = x^2 + y^2. if r=1 then x^2 + y^2 = 1

sin(theta) = x/r = x

cos(theta) = y/r = y when r=1

sin^-1(x) = theta = cos^-1(y)

sin^-1(x) = cos^-1(y)

maybe the algebraic expression you want is x^2 + y^2 = 1

if x and y are positive, then x=+sqr(1-y^2) and y=+sqr(1-x^2)

it's the circle with radius = 1, but with x>0, y>0, it's just 1/4 of the circle, located in Quadrant I

but if cos(0) = 1, then x=1 and y=0 so y^2 = 0 and y=0

(although you have the x and y's switched, so switch them if you want. Then cos(0)=1 becomes y=1)