Find sin(2x), cos(2x), and tan(2x) from the given information. csc(x) = 8, tan(x) < 0

Solution:

Given, cosec (x) = 8

We know, sin (x) = 1/cosec (x) = 1/8

cos²x = 1 - sin²x

cos²x = 1 - (1/8)²

= 1 - 1/64

= (64 - 1)/64

= 63/64

Taking square root,

cos (x) = 3√7/8

sin 2x = 2 sinx cosx

sin 2x = 2(1/8)(3√7/8)

sin 2x = 3√7/32

cos 2x = 2cos²x - 1

= 2(63/64) - 1

= (63/32) - 1

= (63 - 32)/32

cos 2x = 31/32

tan 2x = sin 2x / cos 2x

= (3√7/32)/(31/32)

= 3√7/31

Therefore, sin(2x) = 3√7/32, cos(2x) = 31/32 and tan(2x) = 3√7/31.

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Find sin(2x), cos(2x), and tan(2x) from the given information. csc(x) = 8, tan(x) < 0

Summary:

From the given information, csc(x) = 8, tan(x) < 0 sin(2x) =  3√7/32, cos(2x) = 31/32 and tan(2x) = 3√7/31.