Find the exact value of each expression for the given value of theta. Tan(theta/2) of theta=pi/3

Answer 1

To find the exact value of tan(theta/2) when theta = pi/3, we can use the half-angle identity for tangent which states:

tan(theta/2) = ± √((1 - cos(theta))/(1 + cos(theta)))

First, we need to find the value of cos(theta). Let's start by finding the value of theta for theta = pi/3.

theta = pi/3

Now, let's find the value of cos(pi/3). Since cos(pi/3) represents the x-coordinate of the point on the unit circle, where the terminal side of theta (pi/3) intersects the unit circle:

cos(pi/3) = 1/2

Now, substitute the value of cos(theta) into the half-angle identity:

tan(theta/2) = ± √((1 - cos(theta))/(1 + cos(theta)))
tan(pi/6) = ± √((1 - (1/2))/(1 + (1/2)))

Simplifying further:

= ± √((1/2)/(3/2))
= ± √(1/3)
= ± 1/√3

Therefore, the exact value of tan(theta/2) when theta = pi/3 is ± 1/√3.