Answer: 17/(5sqrt13).
- A (8+3sqrt(13))/(5sqrt(13))
- B (8-3sqrt(13))/(5sqrt(13))
- C 17/(5sqrt13)
- D 6/(5sqrt13)
Correct answer: C. 17/(5sqrt13)
Explanation: For cos-1(4/5): cos=4/5, sin=3/5. For tan-1(2/3): in a right triangle with opposite 2, adjacent 3, hyp sqrt13, so sin=2/sqrt13, cos=3/sqrt13. sin(A+B)=sinAcosB+cosAsinB = (3/5)(3/sqrt13)+(4/5)(2/sqrt13) = (9+8)/(5sqrt13) = 17/(5sqrt13).
The graph of sin⁻¹x is confined to a narrow domain [-1,1] (since sine itself only takes values in that range) and range [-π/2,π/2] (the principal value branch chosen to make sine one-one and invertible there).
Concept context
Restricting trig functions to make them invertible, the principal value branches of sin-inverse, cos-inverse, tan-inverse, and friends, and the key identities relating them.