Use a double or half angle identity to find the exact value of the following expression: if cos x= 4/5 and 270° < x < 360°, find sin 2x

Question

Please help! Thanks

Answer 1

cosx = 4/5 and x is in quad IV

So sinx = - 3/5

sin 2x = 2sinxcosx
= 2(-3/5)(4/5) = -24/25

Answer 2

if tan x = -7/24 and 3π/2 < x <2π find cot x/2

Thanks so much!

Answer 3

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if tanx = -7/24 and x is in IV
then sin x = -7/25 and cosx = 24/25

cot x/2
= cos x/2 / sin x/2

recall cos 2A = 2cos^2 A - 1 or 1 - 2sin^2 A

cos x = 2cos^ x/2 - 1
24/25 + 1 = 2cos^2 x/2 = 49/25
cos^2 x/2 = 49/50
cos x/2 = -7/(5√2) , if x is in IV , then x/2 is in II , therefore negative

cos x = 1 - 2sin^2 x/2
24/25 - 1 = -2sin^2 x/2 = -1/25
sin^2 x/2 = 1/50
sin x/2 = +1/(5√2)

cot x/2 = cos x/2 / sin x/2
= -7/(5√2) / 1/(5√2) = -7

Answer 4

Thank you soo much!