Question

What is the exact value of c{11\pi }{12}" alt="tan\frac{11\pi }{12}" align="absmiddle" class="latex-formula">
$A. -(2-\sqrt{3} )\\B. -(2+\sqrt{3} )\\\\C. 2-\sqrt{3} \\D. 2+\sqrt{3}$

Answer 1

Notice that

11/12 = 1/6 + 3/4

so that

tan(11π/12) = tan(π/6 + 3π/4)

Then recalling that

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

⇒   tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) tan(y))

it follows that

tan(11π/12) = (tan(π/6) + tan(3π/4))/(1 - tan(π/6) tan(3π/4))

tan(11π/12) = (1/√3 - 1)/(1 + 1/√3)

tan(11π/12) = (1 - √3)/(√3 + 1)

tan(11π/12) = - (√3 - 1)²/((√3 + 1) (√3 - 1))

tan(11π/12) = - (4 - 2√3)/2

tan(11π/12) = - (2 - √3) … … … [A]