Final answer:
To find the remaining trigonometric functions of an angle with a given sin(t) and sec(t) conditions, use the Pythagorean identity for trigonometric functions.
Step-by-step explanation:
The subject of this question is the calculation of the remaining trigonometric functions given sin(t) and a condition on sec(t). The value of sin(t) is provided as -1/6 and the condition that sec(t) is less than 0 indicates that the angle t belongs to either the second or third quadrant of the unit circle, where either sine is positive and cosine is negative (second quadrant), or both are negative (third quadrant). Given that sin(t) is negative, t must be in the third quadrant.
To find the remaining trigonometric functions, we must use the Pythagorean identity, cos^2(t) = 1 - sin^2(t). Calculating the cosine value would give us two possible answers (since we are taking the square root), but since sec(t), which is 1/cos(t), is negative, this narrows it down to cos(t) being negative as well. Thus, by substituting the value of sin(t) into the identity, we can find cos(t), and subsequently, the rest of the trigonometric functions such as tan(t), csc(t), and cot(t).
The Laws of Sines and Laws of Cosines are critical for solving problems related to triangles, especially when dealing with non-right triangles. However, they are not directly relevant to this specific problem since it involves trigonometric functions related to a given angle in the trigonometric circle.