Question

Find sin x/2, cos x/2, and tan x/2 from the given information.

[1]. sin x = 5/13, 0° < x < 90°

1). sin x/2 =________.

2). cos x/2 =________.

3). tan x/2 =________.

[Note: Plz help with these 3]

Answer 1

Use Trigonometric Identities :

sin ( x / 2 ) = + OR - sqroot [ ( 1 - cos x ) / 2 ]

cos ( x / 2 ) = + OR - sqroot [ ( 1 + cos x ) / 2 ]


0° < x < 90° meen x lies in Quadrant I

In Quadrant I sin x and cos x are positive so :

sin ( x / 2 ) = sqroot [ ( 1 - cos x ) / 2 ]

cos ( x / 2 ) = sqroot [ ( 1 + cos x ) / 2 ]


If sin x = 5 / 13 then :

cos x = sqroot [ 1 - ( sin x ) ^ 2 ]

cos x = sqroot [ 1 - ( 5 / 13 ) ^ 2 ] =

sqroot [ 1 - 25 / 169 ] =

sqroot [ 169 / 169 - 25 / 169 ] =

sqroot [ 144 / 169 ] = 12 / 13

cos x = 12 / 13


Now :

sin ( x / 2 ) = sqroot [ ( 1 - cos x ) / 2 ]

sin ( x / 2 ) = sqroot [ ( 1 - 12 / 13 ) / 2 ] =

sqroot [ ( 13 / 13 - 12 / 13 ) / 2 ] =

sqroot [ ( 1 / 13 ) / 2 ] =

sqroot [ 1 / ( 2 * 13 ) ] =

sqroot ( 1 / 26 ) = 1 / sqroot ( 26 )

sin ( x / 2 ) = 1 / sqroot ( 26 )


cos ( x / 2 ) = sqroot [ ( 1 + cos x ) / 2 ]

cos ( x / 2 ) = sqroot [ ( 1 + 12 / 13 ) / 2 ] =

sqroot [ ( 13 / 13 + 12 / 13 ) / 2 ] =

sqroot [ ( 25 / 13 ) / 2 ] =

sqroot [ 25 / ( 2 * 13 ) ] =

sqroot ( 25 / 26 ) = 5 / sqroot ( 26 )

cos ( x / 2 ) = 5 / sqroot ( 26 )


tan ( x / 2 ) = sin ( x / 2 ) / cos ( x / 2 )

tan ( x / 2 ) = [ 1 / sqroot ( 26 ) ] / [ 5 / sqroot ( 26 ) ] =

[ 1 * sqroot ( 26 ) ] / [ 5 * sqroot ( 26 ) ] = 1 / 5

tan ( x / 2 ) = 1 / 5