Question

Find the exact value of the following.

Answer 1

1. \[\sec \left[\tan^{-1} \left( -7/3 \right) \right]\]

2. \[\tan \left[ \cos^{-1} \left( -\sqrt{10}/6 \right) \right]\]

Answer 2

you cannot solve exactly unless you know what "quadrant" you are in

Answer 3

but the idea easy enough. draw a triangle and label the "opposite" side 7, the "adjacent" side 3 and find the hypotenuse by pythagoras
\[h=\sqrt{7^2+3^2}=\sqrt{49+9}=\sqrt{58}\]

Answer 4

then for secant use "hypotenuse over adjacent" to get
\[\sec(\theta)=\frac{\sqrt{58}}{3}\] but you dont know if it is plus or minus