Question

What is the value of tangent (t/2) if cosine (t) = 4/5 and t is in quadrant IV?

1) -1/3
2) 1/3
3) -3
4) 3

Answer 1

Using the half-angle identity for tangent and the given cosine value of t, the value of tangent (t/2) is calculated to be -1/3, considering that 't' is in the fourth quadrant where sine is negative.

Step-by-step explanation:

The value of tangent (t/2) can be found using the half-angle identity for tangent, which is tan(t/2) = (1 - cos(t)) / sin(t). First, we are given that cosine (t) = 4/5, and because 't' is in the fourth quadrant, we know that sine (t) will be negative (since sine is negative in the fourth quadrant). Using the Pythagorean identity sin2(t) + cos2(t) = 1, we find that sin2(t) = 1 - (4/5)2 = 1 - 16/25 = 9/25, thus sin(t) = -3/5 (taking the negative root because 't' is in the fourth quadrant). Substituting these values into the half-angle identity, we get tan(t/2) = (1 - 4/5) / (-3/5) = (1/5) / (-3/5) = -1/3.