-pi/6 on the unit circle diagram is the same as 11pi/6. Think about it as going pi/6 counterclockwise, instead of clockwise.
When evaluating sine, think about the y axis. The -pi/6 on the unit circle would horizontally match up with -1/2 on the y axis if you drew a line starting from the intersection of the edge of the circle and -pi/6 that was parallel to the x axis. Thus, sine of -pi/6 is -1/2.
When evaluating cosine, think of the same as sine but with the x axis. The cosine of -pi/6 is √3/2 .
When evaluating tangent, it is useful to look at the numbers outside the parentheses in the chart I attached below. The tangent of -pi/6 is -√3/3.
You can use a calculator to verify.