# Factoring with cosine

1. Nov 11, 2013

### nickb145

1. The problem statement, all variables and given/known data

9cos(2t)=9cos2(t)-4

for all the smallest four positive solutions
2. Relevant equations

3. The attempt at a solution

I've factored it and pulled out 9cos(t) and made that = u
so i have u2-2u-4
That factors to 3.23606 and -1.236

Next i added 9cos(t) back in. 9cos(t)=3.236 and 9cos(t)=-1.236
then solve for the values for the inverse cos-1(3.236/9) and cos-1(-1.236/9)

am i on the right track? I'm not entirely sure if i'm doing it correctly

2. Nov 12, 2013

### Staff: Mentor

No. You are thinking the equation is quadratic in form - it isn't. Use the double-angle identity to rewrite cos(2t) in a different form.

3. Nov 12, 2013

### nickb145

ak ok now i have 8cos2(t)-9sin2(t)+4

is this better?

4. Nov 12, 2013

### Dick

How did you get that??

5. Nov 12, 2013

### nickb145

ok so i used the double angle forumla cos(2t)=cos2(t)-sin2(t)
9(cos2(t)-sin2(t))=cos2(t)-4
distributed the 9 and 9cos2(t)-9sin2(t))=cos2(t)-4

then subtracted cos2 and added 4. 8cos2(t)-9sin2(t)+4

I'm not doing something right...
I'm thinking i need the half angle formula cos2=1+cos(2u)/2

6. Nov 12, 2013

### nickb145

whoops messed something up here

7. Nov 12, 2013

### Dick

The original equation you posted was 9cos(2t)=9cos2(t)-4. What happened to the second 9? Or was that a typo?

8. Nov 12, 2013

### nickb145

typo. the 9cos2(t) cancelled out now i have -9sin2=-4 -> sin2=4/9

9. Nov 12, 2013

### Dick

That's better.

10. Nov 12, 2013

### nickb145

now ive squarerooted it and now have sin(t)=2/3 now to arcsin(2/3)=.7297. That is one solution

( i think)

then i think i subtract 2pi. 2pi-.7297=5.5534 (second solution i think). I add and subtract pi to .7297 because 2pi would make it too big

so pi-.7297=2.4118
pi+.72972=3.871

11. Nov 12, 2013

### Dick

Sounds about right. But your logic is a little fuzzy. sin(t)=(-2/3) is also a solution. Graph y=sin(t), y=2/3 and y=(-2/3) in the range [0,2*pi] to make sure you understand why all of those work.

Last edited: Nov 12, 2013