Question

`cos [ arctan((12)/(5)) - arcsin ((-3)/(5))]`

how would one find the answer to this question?

Answer 1

Answer: -16/65

Explanation:

Drawing the right triangle (as attached) gives us that
`\arctan \left((12)/(5) \right)=\arcsin \left((12)/(13) \right)`

Also,
`-\arcsin \left(-(3)/(5) \right)=\arcsin \left((3)/(5) \right)`

This means our original expression is equal to:

`\cos \left[\arcsin \left((12)/(13) \right)+\arcsin \left((3)/(5) \right) \right]`

Using the cosine addition formula, which states
`\cos(a+b)=\cos a \cos b-\sin a \sin b`, we get this itself is equal to:

`\cos \left(\arcsin \left((12)/(13) \right) \right)\cos \left(\arcsin \left((3)/(5) \right)\right)-\sin \left(\arcsin \left((12)/(13) \right) \right)\sin \left(\arcsin \left((3)/(5) \right)\right)`

Since
`\sin^(2) \theta+\cos^(2) \theta=1`, we know that:

`\sin^(2) \left(\arcsin \left((12)/(13) \right)\right)+\cos^(2) \left(\arcsin \left((12)/(13) \right)\right)=1\\\\(144)/(169) +\cos^(2) \left(\arcsin \left((12)/(13) \right)\right)=1\\\\cos^(2) \left(\arcsin \left((12)/(13) \right)\right)=(25)/(169)\\\\cos \left(\arcsin \left((12)/(13) \right)\right)=(5)/(13)`

Similarly, cos(arcsin(3/5))=4/5.

This means the given expression is equal to:

`\left((5)/(13) \right) \left((4)/(5) \right)-\left((12)/(13) \right) \left((3)/(5) \right)\\\\(20)/(65)-(36)/(65)=\boxed{-(16)/(65)}`