Question 920764: Find the exact value of the expression.
tan(5π/6) − tan(π/6)
/
1 + tan(5π/6)tan(π/6)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you recognize that:
tan(a-b) = tan(a) - tan(b)
------------------------
1 + tan(a) * tan(b)
then this problem becomes easy.
simply replace a with 5pi/6 and b with pi/6 and solve.
you will actually be solving this from tan(a-b) though, and not from the right side of the equation.
when a = 5pi/6 and b = pi/6, then a-b = 4pi/6 which is the same as 2pi/3
you are then solving for tan(2pi/3)
it's best to convert this to degrees if you are not intimately familiar with radians.
2pi/3 * 180 / pi = 120 degrees.
you are looking for tangent of 120 degrees.
that's in the second quadrant.
to make it easier to work with, convert it to the equivalent angle in the first quadrant.
180 - 120 = 60 degrees.
the equivalent angle in the first quadrant is 60 degrees.
this is one of the special triangles that is a 30, 60, 90 degree triangle.
in that triangle.
sin(30) = 1/2
sin(60) = sqrt(3) / 2
cos(30) = sqrt(3) / 2
cos(60) = 1/2
since tan(60) = sin(60) / cos(60), you get sqrt(3) / 2 divided by 1/2.
that results in tan(60) = sqrt(3).
that's in the first quadrant.
your angle is in the second quadrant.
in the second quadrant, the tangent is negative.
since 120 degrees is the equivalent angle in the second quadrant of 60 degrees in the first quadrant, then tan(120) = - sqrt(3).
that's your solution.
tan(2pi/3) = - sqrt(3) because 2pi/3 is the radian equivalent of 120 degrees.
if you had worked this problem in radians, you would have had to recognize that 2pi/3 is in the second quadrant.
you would then have translated that to the first quadrant by taking pi and subtracting 2pi/3 from it to get pi/3 as the equivalent angle in the first quadrant and then recognizing that pi/3 is one of the special triangles with angles of pi/6, pi/3, pi/2.
that's a lot harder to do and most people don't recognize that, so converting to degrees to work with the angle is the way to go unless you are intimately familiar with radians and can work with them directly without translating to degrees first.
the 2 special triangles most often used are 45, 45, 90 and 30, 60, 90.
i confirmed using my calculator that the solution is correct.
i used the calculator to solve the problem directly and then determined that the decimal solution was equivalent to -sqrt(3).
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