anonymous one year ago Find all solutions in the interval [0, 2π). 7 tan3x - 21 tan x = 0

1. zepdrix

Is that $$\large\rm \tan(3x)$$ or $$\large\rm \tan^3x$$ ? :)

2. anonymous

tan ^3 x

3. zepdrix

$\large\rm 7\tan^3x-21\tan x=0$Let's factor some stuff first. Looks like they both have..... a tangent... and a 7, ya?

4. anonymous

7tanx(tan^2x-3)=0

5. zepdrix

Ok good! Now we apply our Zero-Factor Property, setting each individual factor equal to zero, and then solving from there. So our first factor equal to zero is: $$\large\rm 7\tan x=0$$

6. zepdrix

You can divide by 7 to simplify things down a bit: $$\large\rm \tan x=0$$ Understand how to solve for x in this case? :)

7. anonymous

would you plug it into a calculator using tan^-1?

8. zepdrix

I guess you could do that :p it's better just remember some of your special angles. tangent is 0 when the angle x is 0.

9. zepdrix

Oh but I guess umm... they want all of the solutions from 0 to 2pi, so that produces another angle as well.

10. anonymous

oh yeahhhh, so then pi and 0? or would it be 1/pi and 3/pi?

11. zepdrix

sorry got distracted :) um um... yaaaaa, 0 and pi sound good for our first two solutions.

12. zepdrix

Applying the zero-factor property again:$\large\rm \tan^2x-3=0$

13. zepdrix

So ummm... how bout.. add 3. then square root or something, ya?

14. zepdrix

$\large\rm \tan^2=3$

15. zepdrix

$\large\rm \tan x=\pm\sqrt{3}$

16. zepdrix

The positive root 3 will give you 2 angles again. while the negative root 3 will give you 2 more angles! So this solution set is actually giving us 4 angles! kinda crazy.

17. anonymous

uhhhhh... how do you get your solutions from $\pm \sqrt{3}$

18. zepdrix

Hmm you gotta get better with your unit circle missy! :O http://4.bp.blogspot.com/-FMVojhlkcSQ/UInlIKO88oI/AAAAAAAAAC4/-fzMOz6di2Y/s1600/image010.jpg So umm.. i couldn't find a really good picture, but here is one.

19. zepdrix

The coordinate pair lists first the sine value of that angle, and then the cosine. And then outside of the brackets, on the right, is the tangent value of that angle. So if we look in the first quadrant, it looks like the angle pi/3 gives us sqrt(3), ya?

20. anonymous

Ohhhhh! I see now! :P Thank you so much!

21. zepdrix

cool c: