Question 1010969
tan^2(x) = 0 when x ={0, pi, 2pi} as shown below:


<img src = "http://theo.x10hosting.com/2015/121806.jpg" alt="$$$" </>


tan^2(2x) = 0 when 2x = {0, pi, 2pi}.
solve for x and you get x = {0, pi/2, pi, 3pi/2, 2pi} as shown below:


<img src = "http://theo.x10hosting.com/2015/121807.jpg" alt="$$$" </>


tan^2(2x-1) = 0 when (2x-1) = {0, pi, 2pi}
solve for x and you get x = {.5, 2.071, 3.642, 5.212} as shown below:


<img src = "http://theo.x10hosting.com/2015/121808.jpg" alt="$$$" </>


having access to a grahping calculator so you can see what's going on helps a lot.


the calculator i used is at http://www.desmos.com/calculator.


how would you solve this algebraically?


see below for what i did.


you start with 3tan^2(2x-1) = 0


divide both sides of this equation by 3 to get:


tan^2(2x-1) = 0


take the square root of both sides of this equation to get:


tan(2x-1) = 0


this is true if and only if tan^-1(0) = 2x-1


you know that tan is 0 when the angle = 0 or 180 or 360.


in radians, tan is 0 when the angle = 0 or pi or 2pi.


since the angle we are looking for is represented by (2x-1), this means that (2x-1) = 0 or pi or 2pi.


solve for x to get x = .5 or 2.070796327 or 3.641592654


since the formula is tan^2(2x-1) = 0, the frequency is 2 which means the period is pi instead of 2pi.


in order to get results up to 2pi, you need to go through 2 periods.


this means the values of x will repeat every pi radians.


.5 + pi = 3.641592654 + pi = 6.783185307.


2.070796327 + pi = 5.212388981 + pi = 8.35981634


since 2pi = 6.283185307 radians, you only need the values that are between 0 and 6.283185307 radians.


those values are:


.5, 2.070796327, 3.641592654, or 5.212388981 radians.


those are the values shown on the final graph rounded to 3 decimal places.


you should see the following values on the third graph abo ve.


.5, 2.071, 3.642, 5.212.


to test if these values for x are correct, take one of them and follow the original formula to see if it rsults in 0.


for example, tan^2(2x-1) = 0 becomes tan^2(2*3.641592654 - 1) = 0 when x = 3.641592654.


i used my calculator to get tan^2(2*3.641592654-1) = 6.7305616 * 10^-19 which is extremely close to 0.


3.64592654 is a rounded number.


the internally stored number has more decimal places than that.


since 3.6459... is the result of .5 + pi, i'll use that instead to get tan^2(2*(.5+pi)-1) = 0


the difference is in the rounding.