How to find the exact values of sin(u/2), cos(u/2), tan(u/2) using the half angle formulas given sin u=5/13, pi/2<u<pi?

Kai Kerr

Kai Kerr

Answered question

2023-03-09

How to find the exact values of sin ( u 2 ) , cos ( u 2 ) , tan ( u 2 ) using the half angle formulas given sin u = 5 13 , π 2 < u < π ?

Answer & Explanation

planeiaxuf

planeiaxuf

Beginner2023-03-10Added 2 answers

sin u = 5 13 . First, find cos u.
cos 2 u = 1 - sin 2 u = 1 - 25 169 = 144 169 --> cos u = ± 12 13 .
Since u is in Quadrant 2, then, cos u < 0.
cos u = - 12 13 .
To find cos ( u 2 ) , trigonometric identity in use
2 cos 2 ( u 2 ) = 1 + cos u = 1 - 12 13 = 1 13
cos 2 ( u 2 ) = 1 26 --> cos ( u 2 ) = ± 1 26 .
Sin u is in Quadrant 2, then u 2 is in Quadrant 1, and cos ( u 2 ) is positive:
cos ( u 2 ) = 1 26 = 26 26
To find sin ( u 2 ) , apply the trig identity:
sin u = 2 sin ( u 2 ) . cos ( u 2 )
sin ( u 2 ) = sin u 2 cos ( u 2 ) = ( 5 13 ) ( 26 2 ) = 5 26 26
tan ( u 2 ) = sin ( u 2 ) cos ( u 2 ) = ( 5 26 26 ) ( 26 ) = 5

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