Question

Find the exact value of the expression...

sin (sin^-1(2/3) + cos^-1(1/3))

Answer 1

AHHH sorry got busy for a few minutes there >.<
Maybe I can help ya here.

Answer 2

Confused on where to start?

Answer 3

\[\Large\rm \sin\left[\color{orangered}{\sin^{-1}\left(\frac{2}{3}\right)}+\color{royalblue}{\cos^{-1}\left(\frac{1}{3}\right)}\right]\]So like....

Answer 4

You know how the trig functions produce a ratio of sides?
Example:\[\Large\rm \sin\theta=\frac{5}{7}\]We're taking the sine of some angle, and it gives us this ratio of opposite to hypotenuse.

Answer 5

Inverse trig functions do the reverse.
Example:\[\Large\rm \sin^{-1}\left(\frac{17}{4}\right)=\phi\]We're taking the inverse sine of some ratio of sides, and it spits out an angle.

Answer 6

So that's how we want to deal with this problem.
Think of the `orange` and `blue` and two different angles, and we're going to do some fancy stuff with them.

Answer 7

So our problem really looks something like this:\[\Large\rm \sin\left[\color{orangered}{\theta}+\color{royalblue}{\phi}\right]\]Where,\[\Large\rm \color{orangered}{\theta=\sin^{-1}\left(\frac{2}{3}\right)}\]\[\Large\rm \color{royalblue}{\phi=\cos^{-1}\left(\frac{1}{3}\right)}\]

Answer 8

We can rewrite these in terms of the normal trig functions,\[\Large\rm \color{orangered}{\sin \theta=\frac{2}{3}}\]\[\Large\rm \color{royalblue}{\cos \phi =\frac{1}{3}}\]

Answer 9

And graph them on `separate` triangles using the side relations which are given.

Answer 10

Do you see how we can use the 2/3 to label this triangle?
Where you at Abbot, you around? :D