Question

determine whether the positive or negative square root should be selected
\tan 50 ^ { circ } = pm sqrt { \frac { 1 - cos 100 ^ { circ } } { 1 + cos 100 ^ { circ } } }
since ( 50 ^ { circ } ) is in quadrant the square root should be selected

Explanation:

Step1: Recall the sign of tangent in different quadrants

The formula for \(\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{1+\cos\alpha}}\). Here \(\alpha = 100^{\circ}\), so \(\frac{\alpha}{2}=50^{\circ}\).
The tangent function \(y = \tan x\) has the sign: positive in the first and third quadrants (\(\tan x=\frac{\sin x}{\cos x}\), and \(\sin x\) and \(\cos x\) have the same sign in the first and third quadrants), negative in the second and fourth quadrants (\(\sin x\) and \(\cos x\) have opposite signs in the second and fourth quadrants).

Step2: Determine the quadrant of \(50^{\circ}\)

Since \(0^{\circ}<50^{\circ}<90^{\circ}\), \(50^{\circ}\) is in the first quadrant.
In the first quadrant, \(\tan50^{\circ}>0\).

Answer:

Since \(50^{\circ}\) is in quadrant \(I\), the \(positive\) square root should be selected.