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

Question

csc x = 6, tan x < 0?

Answer 1

csc(x) = 6

Reciprocal identity of cscx is sinx.

1/sinx = 6

sinx = 1/6

tanx < 0

sinx is positive and tanx is negative in second quadrant.

From the basic trigonometric ratios, sin(θ) = (opposite side/hypotenuse)

From Pythagorean theorem,

Adjacent side = √[(hypotenuse)²-(opposite side)²]

= √(36 - 1) = √35

cosx = (adjacent side/hypotenuse)

= -√35/6

Sin(2x) = 2 sinx cosx

= 2 (1/6)(-√35/6)

Sin(2x) = - √35/18

cos(2x) = cos²x - sin²x

= (-√35/6)² - (1/6)²= (35/36) - (1/36) = 34/36

cos(2x) = 17/18

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

= (- √35/18)/(34/36)

tan(2x) = (- √35)/17.