## animalsavior94 Group Title Find all solutions in the interval [0, 2pi). give exact values and show all algebra. 2cos^2 X +3sinX-3=0 3 years ago 3 years ago

1. satellite73 Group Title

trick is to rewrite $cos^2(x)= 1-sin^2(x)$

2. satellite73 Group Title

then get a quadratic equation in $sin(x)$

3. satellite73 Group Title

$2cos^2(x)+3sin(x)-3=0$ $2(1-sin^2(x))+3sin(x)-3=0$ $2-2sin^2(x)+3sin(x)-3=0$ $-2sin^2(x)+3sin(x)-1=0$ $2sin^2(x)-3sin(x)-1=0$

4. satellite73 Group Title

just solve as a quadratic. if it is easier to see replace sin(x) by z to get $2z^2-3z+1=0$ $(2z-1)(z-1)=0$ $z=\frac{1}{2}$ or $z=1$

5. satellite73 Group Title

then replace z by sin(x) and solve for x $sin(x)=1$ $x=\frac{\pi}{2}$

6. satellite73 Group Title

$sin(x)=\frac{1}{2}$ $x=\frac{\pi}{6}$ or $x=\frac{5\pi}{6}$

7. satellite73 Group Title

ooops second answer is wrong sorry

8. satellite73 Group Title

$sin(x)=\frac{1}{2}$ $x=\frac{11\pi}{6}$

9. satellite73 Group Title

k?

10. animalsavior94 Group Title

so x=11pi/6?

11. satellite73 Group Title

there are two places on the interval from 0 to 2pi where sine is 1/2

12. satellite73 Group Title

one is at $\frac{\pi}{6}$ and the other is at $\frac{11\pi}{6}$ so all together you have 3 answers. those two and also $\frac{\pi}{2}$ where sine is 1

13. animalsavior94 Group Title

ohh i see thank you!

14. satellite73 Group Title

welcome!

15. satellite73 Group Title

any more?

16. animalsavior94 Group Title

huh?

17. satellite73 Group Title

any more of these to do?

18. animalsavior94 Group Title

well i just put some up

19. Gabysolis49 Group Title

Does anyone happen to know 2cos^2 x+sinx-1=0 solve for x on [0,2pi)

20. Gabysolis49 Group Title

@satellite73