## anonymous one year ago Find all solutions to the equation. 7 sin2x - 14 sin x + 2 = -5

1. mathmate

Not sure if you want to solve: $$7\sin^2(x)-14\sin(x)+2=-5$$ or $$7\sin(2x)-14\sin(x)+2=-5$$ In either case, factor out the common factor 7 to simplify things a little.

2. anonymous

sorry the first one

3. anonymous

But I can't factor out 7 because of the 2

4. phi

first put it in standard form by adding +5 to both sides $7\sin^2(x)-14\sin(x)+2=-5 \\ 7\sin^2(x)-14\sin(x)+7=0$

5. Astrophysics

You can always let sinx = x then factor normally and plug back sinx to find the solution

6. Astrophysics

$7x^2-14x+2=5$

7. anonymous

I'm sorry. Are you saying all the possible solutions are x=±7 x=±14 and x=±2?

8. phi

$7\sin^2(x)-14\sin(x)+7=0$ divide both sides (and all terms) by 7. you get $\sin^2(x)-2\sin(x)+1=0$ to avoid confusion, let $$y= \sin x$$ and write this as $y^2 -2 y +1= 0$ can you find the values for y ?

9. anonymous

I'm so confused. My brother said y=(sqrt-.5)+y

10. phi

you can factor y^2 -2 y + 1 =0 to get (y-1)(y-1)=0 now solve for y-1= 0

11. phi

you get y=1 remember y= sin x so sin x =1 what "x" values have sin x = 1?

12. Astrophysics

Letting x = sinx we get $7x^2-14x+2=-5 \implies 7x^2-14x+7 =0$ divide both sides by 7 to simplify, we get $x^2-2x+1=0$ factoring we $(x-1)^2=0 \implies (x-1)(x-1) = 0$ Now we sub sinx back in to get $(sinx-1)^2 = 0 \implies (sinx-1)(sinx-1) = 0$ so solving for just sinx -1 = 0, we get sinx = 1

13. anonymous

So there's only one value?

14. phi

sin goes forever. between 0 and 2pi (or 0 and 360 degrees) there is only pi/2 (90 degrees) but you can add 2pi to that answer and you will get another value that works people would write the answer as $x = \frac{\pi}{2} + k \cdot 2 \pi, \text{ k any integer}$

15. anonymous

Thank you so much guys!