Find the exact value of this inverse function

1. Apr 21, 2010

1irishman

1. The problem statement, all variables and given/known data
Find the exact value of cos(2arcsin(-1/8))

2. Relevant equations
make use of the double angle formula

3. The attempt at a solution
let arcsin(-1/8)=theta
then sin theta= -1/8

a=sqrt63=3sqrt7

2. Apr 21, 2010

Staff: Mentor

So now evaluate cos(2theta).
Where does a come in?

3. Apr 22, 2010

1irishman

'a' was a side i named

4. Apr 22, 2010

Staff: Mentor

OK. Still, you need to evaluate cos(2theta). One identity is cos(2theta) = cos^2(theta) - sin^2(theta). This identity appears in two other forms, one of which will be useful for this problem.

5. Apr 23, 2010

1irishman

i'm still lost...

6. Apr 24, 2010

LCKurtz

Draw a picture of the angle θ and label the sides of a little triangle appropriately to have a sine of -1/8. You can read all six trig functions off that triangle and should be able to calculate cos(2θ) from there.

7. Apr 24, 2010

HallsofIvy

Staff Emeritus
Since you were told "make use of the double angle formula", $cos(2\theta)= cos^2(\theta)- sin^2(\theta)= 1- 2sin^2(\theta)$.

You know that $sin(\theta)= \frac{1}{8}$.

8. Apr 26, 2010

1irishman

what does cos^2 equal?

9. Apr 26, 2010

LCKurtz

cos2(θ) is just an abbreviation for (cos(θ))2.

10. Apr 27, 2010

Mentallic

Once you draw your right-triangle and label an angle $\theta$ then since sin$\theta$ = -1/8, you know what your opposite and hypotenuse sides are, then you can use pythagoras' theorem to find the 3rd side. This will give you cos$\theta$.

11. Apr 27, 2010

HallsofIvy

Staff Emeritus
cos(2x)$= cos^2(x)- sin^2(x)$$= (1- sin^2(x))- sin^2(x)= 1- 2sin^2(x)$

If x= arcsin(-1/8) what is sin(x)?

I just notice that I had said this same thing several days ago!

Last edited: Apr 28, 2010
12. Apr 27, 2010

1irishman

sin x is -1/8

13. Apr 27, 2010

Mentallic

Right, so what is cos(2x) given that cos(2x)=cos^2(x)-sin^2(x) ?

14. Apr 28, 2010

HallsofIvy

Staff Emeritus
Which, as I pointed out in the post that Irishman's post was in response to (and, I just noticed, several days ago!) , is equal to $1- 2sin^2(x)$.

Irishman, if sin(x)= -1/8, what is $sin^2(x)$? What is $1- 2sin^2(x)$?

15. Apr 28, 2010

Mentallic

Yes, I know, this identity has been mentioned at least 3 times in this thread now. It just seems like 1irishman is only considering and responding to the most recent reply in this thread each time he logs back on, so I felt the need to hold his hand as we guide him through the process...