trigonometic?

Question

please explain why Tan alpha = sqrt(3) is (pi/3) + (pi)(N)

Answer 1

To see that tan(pi/3) = sqrt(3), draw an equilateral triangle (all angles pi/3) and bisect one of the angles. The two resulting triangles are congruent, and the length of the bisector is:
sqrt(2^2 - 1^2) = sqrt(3).

Angles in the first and third quadrants have positive tangents, whereas those in the second and fourth quadrants do not.

theta + n*pi is the general formula which equates all angles sharing with the same tangent as any angle theta.

Put theta = pi/3 in that formula to get the result you require.

Answer 2

Whenever yo see the square root of 3 you should thing of a 30-60-90 triangle. In such a triangle, the hypotenuse is 2 the shorter leg is 1 and the longer leg is the square root of 3.

The tangent of 60 degrees = tan pi/3 =sqrt(3)
The tangent is also positive in the 3rd quadrant at 240 degrees = pi/3+pi
This repeats every increment of pi so we have pi/3+pi(n), where n= 1,2,3,4 .....n

Answer 3

If tan α = √3

then α = π/3 +nπ

This is because arctan (√3) is π/3 in the first quadrant but tanα is periodic and repeats every π radians.

So if n is any whole number then π/3 +nπ must map all of the possible solutions of tan α = √3.