Find the exact value of the expression when it is defined.

Question

sin [sin^-115/13 - cos^-1(-3/5)]

Answer 1

We should use the difference formula for sine: sin(A - B) = sinAcosB - sinBcosA

Also, I will proceed on the assumption that there is a typo and the question is about sin^-1(5/13) (otherwise the entire expression is undefined since sin^-1(15/13) is undefined).

We can use a diagram of a 5-12-13 right triangle to see that cos(sin^-1(5/13)) = 12/13 (The fundamental trig identity, sin²Θ + cos²Θ = 1 also works to calculate that.)

We can use a diagram of a 3-4-5 triangle to show that sin(cos^-1(-3/5)) = 4/5 (this is positive since cos^-1 of a negative ratio is a QII angle and sine is + in QII).

Finally, we have ...

sin(sin^-1(5/13) - cos^-1(-3/5))

= sin(sin^-1(5/13))cos(cos^-1(-3/5)) - sin(cos^-1(-3/5))cos(sin^-1(5/13))

= 5/13·(-3/5) - 4/5·12/13

= -15/65 - 48/65

= - 63/65