How would I find cos(u/2) if sec u =13.465 and u is in Quadrant-IV (assume that 0<u<2π) ?
cos(u/2)=___
these identities are helpful
cos(u) = 1 / sec(u)
cos(u/2) = ± √{[1 + cos(u)] / 2}
cos(u/2) = ± √({1 + [1 / sec(u)]} / 2)
To find cos(u/2) given sec(u) = 13.465 and u is in Quadrant IV, we can use the trigonometric identities to relate sec(u) to cos(u).
Recall that sec(u) is the reciprocal of cos(u), thus:
sec(u) = 1/cos(u)
Rearranging the equation, we get:
cos(u) = 1/sec(u)
Now, substitute the given value of sec(u) into the equation:
cos(u) = 1/13.465
Next, to find cos(u/2), we will use the half-angle identity for cosine:
cos(u/2) = sqrt((1 + cos(u)) / 2)
First, substitute the value of cos(u) that we just found:
cos(u/2) = sqrt((1 + 1/13.465) / 2)
Simplifying further:
cos(u/2) = sqrt((13.465 + 1)/ (13.465 * 2))
cos(u/2) = sqrt(14.465/26.93)
cos(u/2) = sqrt(0.5370)
Therefore, cos(u/2) = ~0.7328 (approximately)