(d/dx)sin(x) = cos(x).

Angles in degrees require a multiplier of 180/pi which makes the formulae more cumbersome. It is also customary to assume that when an angle is expressed without any units, it is in radians.

1st. example.

Let x = cos^(-1)(sin(-2)).

-2 radians are equal approximately to -115deg, making this a 3rd. quadrant angle with a negative sine.

As a positive angle, it is 2pi – 2.

The principal value of cos^(-1) lies in the range [0, pi], and so for a negative value yields a 2nd quadrant angle.

If a and b are both acute angles such that

a = cos^(-1)(sin(b)),

then cos(a) = sin(b),

and a + b = pi/2.

x is therefore a 2nd quadrant angle such that the radius vector is perpendicular to that for 2pi – 2.

Thus:

x = 2pi – 2 – pi/2

x = 3pi/2 – 2.

2nd. example.

Let x = sin^(-1)(cos(2)).

2 radians is a 2nd quadrant angle, and its reference angle is pi – 2.

The reference angle for the sin^(-1) is pi/2 – (pi – 2) = 2 – pi/2

Principal value of sin^(-1) for a -ve value is an angle in [0, -pi/2].

x = pi/2 – 2.