Question

How do you simplify `sin[cos^-1( - sqrt5 / 5 ) + tan^-1 ( - 1 / 3) ]`?

Answer 1

`=+-0.7 sqrt 2 and +-1/sqrt 2=+-0.98995 and +-0.70711`, nearly.

Explanation:

Let `a = cos^(-1)(-sqrt 5/5)`. Then `cos a = -sqrt 5/5=-1/sqrt 5<0`. so, a is in the 2nd quadrant or in the 4th. Accordingly, `sin a = +-2/sqrt 5`.

Let `b = tan^(-1)(-1/3)`. Then `tan b = -1/3<0`. So, b is in the 2nd quadrant or in the 4th. Accordingly, `sin b = +-1/sqrt 10 and cos b = 3/sqrt 10`. Also, sin b and cos b have opposite signs.

Now, the given expression is

sin ( a + b ) = sin a cos b + cos a sin b

`=(+-2/sqrt 5)(+-3/sqrt 10)-(-1/sqrt 5)(+-1/sqrt 10)`

`=+-6/sqrt 50` `-` or + `1/sqrt 50`

`=(sqrt 2/10)(+-6+-1)`

`=+-0.7 sqrt 2 and +-1/ sqrt 2`

In each case, the angles a and b can be obtained separately. .