Final answer:
For t = 0, the values of the trigonometric functions are sin(0) = 0, cos(0) = 1, tan(0) = 0, csc(0) is not defined, sec(0) = 1, and cot(0) is not defined.
Step-by-step explanation:
To evaluate the value of each of the six trigonometric functions for t = 0, we recall the unit circle definition of these functions.
In a unit circle, t represents the angle in radians with respect to the positive x-axis. For t = 0, we are looking at the point (1,0) on the unit circle.
The six trigonometric functions evaluated at this point are:
- Sine (sin(0)) = 0
- Cosine (cos(0)) = 1
- Tangent (tan(0)) = 0
- Cosecant (csc(0)) = Not defined (since sin(0) = 0 and csc is 1/sin)
- Secant (sec(0)) = 1 (since cos(0) = 1 and sec is 1/cos)
- Cotangent (cot(0)) = Not defined (since tan(0) = 0 and cot is 1/tan)
The values of sine and tangent are zero because their respective y and x component ratios define them at this angle, while cosine and secant are one due to their relation to the x component, which is at its maximum on the unit circle at this angle. Cosecant and cotangent are not defined because they are the reciprocals of sine and tangent respectively, and division by zero is undefined.