Final answer:
The trigonometric function values for point P(-sqrt(3)/2, -1/2) are sin t = -1/2, cos t = -sqrt(3)/2, tan t = 1/sqrt(3), csc t = -2, sec t = -2/sqrt(3), and cot t = sqrt(3).
Step-by-step explanation:
In order to find the values of the trigonometric functions for the point P = (-Square root 3/2, -1/2) on the unit circle that corresponds to the real number t, we can use the coordinates of point P, which represent the cosine and sine of the angle t, respectively.
sin t = -1/2
cos t = -sqrt(3)/2
tan t = (sin t)/(cos t) = 1/sqrt(3)
csc t = 1/(sin t) = -2
sec t = 1/(cos t) = -2/sqrt(3)
cot t = 1/(tan t) = sqrt(3)
The sine (sin t) and cosine (cos t) of angle t are given directly by the y-coordinate and x-coordinate of point P on the unit circle, respectively. The tangent (tan t) is the ratio of sine to cosine. The cosecant (csc t), secant (sec t), and cotangent (cot t) are the reciprocals of sine, cosine, and tangent, respectively. Because the point P lies in the third quadrant of the unit circle, where both x and y are negative, the values of sine and cosine are also negative. When taking reciprocals for secant and cosecant, the negative signs are preserved. The values for tangent and cotangent are positive since the negatives cancel out, and they are simplified by using the property that sqrt(3)/2 is approximately 0.866.