Question

How do you simplify `sin[Tan^-1 (-2/5) + Cos^-1 (3/5)]`?

Answer 1

`14/(5sqrt 29)`, against principal values of inverse functions, in the expression. General values are `+-14/(5sqrt 29) and +-26/(5sqrt 29)`.

Explanation:

Let `a = tan^(-1)(-2/5)`. Then, `tan a =-2/5 < 0`.

So, principal a is in the 4th quadrant.

And so, sin is negative and cos is positive.

Thus, in this case, `cos a =5/sqrt 29 and sin a = -2/sqrt 29`.

Let `b = cos^(-1)(3/5)`. Then, #cos b =3/5 > 0.

So, principal b is in the 1st quadrant.

And so, sin is positive..

Thus, in this case, sin b = 4/5.

The given expression = sin (a + b)=sin a cos b+cos a sin b

`=(-2/sqrt29)(3/5)+(5/sqrt 29)(4/5)`

`=14/(5sqrt 29)`.

In general, a is in either 4th quadrant or in the 2nd and b is in either

1st or the 2nd. So, sin a and cos a can take both signs.

ln general, b is in either 1st or 4th. So, sin b can take both signs.

Accordingly, the general values are

`+-14/(5sqrt 29) and +-26/(5sqrt 20)`.