Final answer:
The coordinates of point P on the unit circle corresponding to t = 5π/6 are (-√3/2, 1/2). The six trigonometric functions of t are sin(t) = 1/2, cos(t) = -√3/2, tan(t) = -√3/3, csc(t) = 2, sec(t) = -2√3/3, and cot(t) = -√3.
Step-by-step explanation:
Given that t = 5π/6, we can determine the coordinates of point P on the unit circle using the trigonometric functions. The unit circle has a radius of 1, so the x-coordinate of P can be found using the cosine function and the y-coordinate can be found using the sine function.
The x-coordinate of P, denoted as x, is given by x = cos(t). Plugging in t = 5π/6, we have x = cos(5π/6).
The y-coordinate of P, denoted as y, is given by y = sin(t). Plugging in t = 5π/6, we have y = sin(5π/6).
Using the unit circle, we can find that x = -√3/2 and y = 1/2.
Now, we can evaluate the six trigonometric functions of t using the coordinates of P.
The six trigonometric functions are:
- Sine (sin): sin(t) = y = 1/2
- Cosine (cos): cos(t) = x = -√3/2
- Tangent (tan): tan(t) = y/x = (1/2)/(-√3/2) = -1/√3 = -√3/3
- Cosecant (csc): csc(t) = 1/sin(t) = 1/(1/2) = 2
- Secant (sec): sec(t) = 1/cos(t) = 1/(-√3/2) = -2/√3 = -2√3/3
- Cotangent (cot): cot(t) = 1/tan(t) = 1/(-√3/3) = -3/√3 = -√3