Question

How do you find the exact value of `cos[2 arcsin (-3/5) - arctan (5/12)]`?

Answer 1

Possible values are `+-0.1108` or `+-0.6277`

Explanation:

`arcsin(−3/5)=x` means `sinx=(-3/5)=-0.6`.

As `sinx=0.6` for `x=36.87^o` and sine is negative in third and fourth quadrant, `x=180^o+36.87^o` or `216.87^o` and `x=360^o-36.87^o=323.13^o`.

`arctan(5/12)=x` means `tanx=(5/12)`.

As `tanx=5/12` for `x=22.62^o` and tan is positive in first and third quadrant, `x=22.62^o` or `x=180^o +22.62^o`.or `202.62^o`.

Hence `cos{2arcsin(−3/5)-arctan(5/12)]=cos[2xx216.87^o-22.62^o]=cos411.12^o=cos51.12^o=0.6277` or

`cos{2arcsin(−3/5)-arctan(5/12)]=cos[2xx216.87^o-202.62^o]=cos411.12^o=cos231.12^o=-0.6277` or

`cos{2arcsin(−3/5)-arctan(5/12)]=cos[2xx323.13^o-22.62^o]=cos623.64^o=-0.1108` or

`cos{2arcsin(−3/5)-arctan(5/12)]=cos[2xx323.13^o-202.62^o]=cos443.64^o=0.1108`

Answer 2

Values in exactitude are `+-12/125` and `+-68/125`.

Explanation:

Let `A = arc sin (-3/5)` and `B=arc tan (5/12)`.
Then, `sin A = -3/5, cos A = +-4/5`.

`cos 2A = 1-2 sin^2A=7/25`

`sin 2A=2 sin A cos A=-24/25 and 24/25`, respectively..

Also, `tan B=5/12, (sin B =5/13, cos B=12/13) and (sin B =-5/13, cos B=-12/13)`

The given expression is

cos(2A-B) = cos 2A cos B+sin 2A sin B

= `+-(84+-120)/375`.

Note that both sin B and cos B have the same sign + or `-`.