How do you express sin(sin^(-1)(x)+cos^(-1)(y)) without trigonometric functions?
1 Answer
Explanation:
Let
Then:
-pi/2 <= alpha <= pi/2" " so" "cos alpha >= 0
0 <= beta <= pi" " so" "sin beta >= 0
By Pythagoras:
cos^2 theta + sin^2 theta = 1
Hence we have:
sin(alpha) = sin(sin^(-1)(x)) = x
cos(alpha) = sqrt(1-sin^2(alpha)) = sqrt(1-x^2)
cos(beta) = cos(cos^(-1)(y)) = y
sin(beta) = sqrt(1-cos^2(beta)) = sqrt(1-y^2)
Noting that we can use the non-negative square root in both these cases from our prior observation that
Then using the sum formula for
sin(sin^(-1)(x)+cos^(-1)(y)) = sin(alpha+beta)
color(white)(sin(sin^(-1)(x)+cos^(-1)(y))) = sin(alpha)cos(beta)+sin(beta)cos(alpha)
color(white)(sin(sin^(-1)(x)+cos^(-1)(y))) = xy+sqrt(1-y^2)sqrt(1-x^2)
