tan-1 (-sq rt 3/3)

Question

equals pi/6 but how did it follow

Answer 1

Draw an equilateral triangle and divide it into two right triangles by bisecting a vertex angle. This will also be a perpendicular bisector. Now we have two triangles with angles 60, 30 and 90 degrees. If the equilateral triangle has sides of unit length, then the bisected side will be split into two lengths of ½ each. The perpendicular has a length of √3/2 (Pythagoras). The tangent of the bisected vertex angle is ½/(√3/2)=1/√3. Multiply top and bottom by √3 and we get √3/3. The bisected angle is 60/2=30 degrees or π/6. So tan(π/6)=√3/3, or π/6=tan⁻¹(√3/3).