Re: Algebraic Expressions
NoAsherelol said:
Wirte and Algebraic expression for
Cos(Arccos x - Arcsin X)
This one is a toughy, ok this is what i came up with so far....
Cos( (1/cos x)-(1/sin x)
Cos( Sec x - Csc X)
(Cos * !/ Cos x) - Csc x
1/x -(csc x)
is this correct
I think you're getting your inverse functions mixed up with your reciprocal functions.
cos(arccos x - arcsin x) = cos(arccos x) - cos(arcsin x)
Now, cos(arccos x) = x, so now we have:
x - cos(arcsin x)
Now let's find cos(arcsin x):
Let z = cos ( arcsin x ) and y = arcsin x so that z = cos y. Since y = arcsin x is equivalent to sin y = x, with -1 < x < 1 and - pi / 2 <= y <= pi / 2
y = arcsin x may also be written as
sin y = x with - pi / 2 <= y <= pi / 2
Also
sin^2 y + cos^2 y = 1
Substitute sin y by x and solve for cos y to obtain
cos y = +/- sqrt (1 - x^2)
But - pi / 2 <= y <= pi / 2 so that cos y is positive
z = cos y = cos(arcsin x) = sqrt (1 - x^2)
Final answer: cos(arccos x - arcsin x) = x - sqrt(1-x^2)
