Find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas.
sec (u) = -5/2 π/2 < u < π

Solution:

Given that:

sec (u) = -5/2

cos (u) = -2/5

We know the trigonometric laws:

2cos²(u/2) = 1 + cosu

2sin²(u/2) = 1 - cosu

2cos²(u/2) = 1 - 2/5 = 3/5

cos²(u/2) = 3/10

cos(u/2) = √3/10

Since u is in Quadrant I, then cos(u/2) is positive.

2sin²(u/2) = 1 + 2/5 = 7/5

sin²(u/2) = 7/10 

sin(u/2) = √7/10 

Since u is in Quadrant I, then sin(u/2) is positive.

tan(u/2) = sin/cos =√7/3

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Find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas. sec (u) = -5/2 π/2 < u < π

Summary:

The exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas. sec (u) = -5/2 π/2 < u < π are √7/10 , √3/10, √7/3 respectively.