Find sin 2x, cos 2x, and tan 2x from the given infomation
csc x=4, tan x < 0
To find sin 2x, cos 2x, and tan 2x, we first need to find the values of sin x, cos x, and tan x.
Given that csc x = 4, we can find sin x by taking the reciprocal of csc x:
sin x = 1 / csc x
sin x = 1 / 4
Since tan x < 0, we know that tan x is negative. We also know that tan x is equal to sin x / cos x, so we can determine the sign of cos x. Since tan x = sin x / cos x and tan x < 0, we have sin x < 0 and cos x > 0.
Now we have the values of sin x, cos x, and tan x:
sin x = 1/4
cos x = √(1 - sin^2 x) = √(1 - (1/4)^2) = √(1 - 1/16) = √(15/16) = √15/4
tan x = sin x / cos x = (1/4) / (√15/4) = 1/√15
To find sin 2x, we use the double-angle formula: sin 2x = 2sin x cos x
sin 2x = 2(1/4)(√15/4) = √15/8
To find cos 2x, we again use the double-angle formula: cos 2x = cos^2 x - sin^2 x
cos 2x = (√15/4)^2 - (1/4)^2 = 15/16 - 1/16 = 14/16 = 7/8
Finally, to find tan 2x, we use the identity: tan 2x = (2tan x) / (1 - tan^2 x)
tan 2x = (2(1/√15)) / (1 - (1/√15)^2) = (2/√15) / (1 - 1/15)
tan 2x = (2/√15) / (14/15) = (2/√15) * (15/14) = 2/(√15 * 14) = 2√15 / 14√15 = 2/14 = 1/7
Therefore, sin 2x = √15/8, cos 2x = 7/8, and tan 2x = 1/7.