solve tan(x/2)=sqrt3/3

Question

solve on the interval [0,2(pie)).  The square root is just on the top 3 not the whole fraction.

Answer 1

The √3/3 is another way of writing (after the denominator has been rationalized)  1/√3

 

Drawing a 30-60-90 triangle, you will see that the tan (opposite/adjacent) of the 30 degree angle is 1/√3

 

Since your equation reads tan (x/2), in order to get 30 degrees, x must equal 60. 

 

Since you need the answer over a 360 degree span (0, 2pi), you need to ask yourself "in which quadrants is tan a positive value?"  quadrant 1 and 3

 

then ask yourself, what is the angle measure of a 30 degree angle placed in quadrant 3?  210 degrees

 

the values for both answers must be doubled because your equation states tan (x/2).

 

therefore 60 degrees is one answer

 

the other answer of 420 degrees is outside of the (0, 2pi) interval

 

if you need your answer in radians, convert 60 degrees by multiplying it by the π/180 which is π/3

 

I explained it in this fashion because I personally find it easier to think in degrees rather than radians. I then convert to radians at the end if that is the units requested in the answer. 

 

Also this is the way my phone prints the pi symbol.  π

 

 

Answer 2

tan=sin/cos

tangent is positive in the 1st and 3rd quadrants on the unit circle

look on the unit circle

look at ∏/6 or 30º

sin=1/2 and cos=√3/3

sin/cos=(1/2)/(√3/2)=(1/2)(2/√3)=1/√3=√3/3

x/2=30

x=60º

look at 7∏/6 or 210º

sin=-1/2 and cos=-√3/3

when you divide a negative by a negative you get a positive

sin/cos=√3/3 again because you are dividing with the same numbers only they are both negative

x/2=210

x=420º which exceeds 360º