Find the values of other five trigonometric functions if tan x = -5/12, and x lies in the second quadrant.

Sure, let's solve this step by step.

Step 1: We know that tan x = -5/12. The negative sign indicates that x is in the second quadrant where tan is negative.

Step 2: We also know that tan x = sin x / cos x. So, we can say that sin x = -5 and cos x = 12 (considering a right triangle where the opposite side is -5 and the adjacent side is 12).

Step 3: To find the hypotenuse (r) of this right triangle, we can use the Pythagorean theorem: r = sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13.

Step 4: Now, we can find the values of the other trigonometric functions:

• sin x = -5/13 (opposite/hypotenuse)
• cos x = 12/13 (adjacent/hypotenuse)
• csc x = 1/sin x = -13/5
• sec x = 1/cos x = 13/12
• cot x = 1/tan x = -12/5

So, the values of the six trigonometric functions are: sin x = -5/13, cos x = 12/13, tan x = -5/12, csc x = -13/5, sec x = 13/12, and cot x = -12/5.