Final answer:
To find the values of the other five trigonometric functions when cos x = -1/2 and x lies in the third quadrant, we can use the Pythagorean identity and the definitions of the trigonometric functions to determine the values of sin x, tan x, csc x, sec x, and cot x.
Step-by-step explanation:
To find the values of the other five trigonometric functions, we need to know the value of sine (sin x), tangent (tan x), cosecant (csc x), secant (sec x), and cotangent (cot x) when the cosine (cos x) is given as -1/2 and x lies in the third quadrant.
In the third quadrant, the adjacent side is negative and the opposite side is positive. Since cosine is negative in this quadrant, we know that the adjacent side is negative. Let's use the Pythagorean identity to find the value of the opposite side:
sin x = ±√(1 - cos² x) = ±√(1 - (-1/2)²) = ±√(1 - 1/4) = ±√(3/4) = ±√3/2
Now, we can find the values of the other trigonometric functions:
tan x = sin x / cos x = ±(√3/2)/(-1/2) = ∓√3
csc x = 1/sin x = 1/(±√3/2) = ∓2/√3 = ∓2√3/3
sec x = 1/cos x = 1/(-1/2) = -2
cot x = 1/tan x = 1/(∓√3) = ∓1/√3 = ±√3/3