Question

Find the exact value of cos(2cos^-1 (4/5))

Answer 1

Did you draw a right triangle and label the adjacent side 4 and the hypotenuse 5? That would be a great place to start.

Answer 2

Yes, I did. But now I'm stuck. It's a 3-4-5 triangle. I don't know how to incorporate the double angle formulas.

Answer 3

Have you forgotten your identities?

\(\cos(2x) = 2\cos^{2}(x) - 1 = 1 - 2\sin^{2}(x) = \cos^{2}(x) - sin^{2}(x)\)

Remember one thing, though, QUADRANT. I'll leave it at that and see where you land.

Answer 4

Yes, I know the double angle formulas. But I'm confused because what am I supposed to do with the (2cos^-1(4/5)) part. What does the 2 in front of the cos mean?

Answer 5

Spend more time thinking on the clue and less time telling yourself why it isn't a clue.

It doesn't do any good at all to "know" the double angle formula if you do not see the opportunity for its use. This is your chance.

\(\cos(\cos^{-1}(x)) = x\) -- Right? (For suitable x, anyway.)

And the Double Angle?
\(\cos(2\cos^{-1}(x)) = 2[\cos(\cos^{-1}(x))]^{2} - 1\) -- That stuff in the square brackets should look familiar.