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How do you find all six trigonometric function of #theta# if the point #(0,-5)# is on the terminal side of #theta#?

Answer 1

#sin theta = 0, cos theta = -1#
#csc theta = "undefined"; sec theta = -1#
#tan theta = 0, cot theta = "undefined"#

When you place the point #(0, -5)# on a coordinate plane, the angle from the x-axis to the point is #theta = 180^@#counterclockwise.

#sin 180^@ = 0#, which is the #y# distance from the point to the #x#-axis.

#cos 180^@ = -1# from a trig. circle.

So #rcos theta = -1#

Using the trig. function definitions, we can calculate the remaining amount from these two trig. values:

#csc theta = 1/(sin theta) = 1/0 = "undefined"#

#sec theta = 1/(cos theta) = -1 #

#tan theta = (sin theta)/(cos theta) = 0/-1 = 0#

# cot theta = 1/(tan theta) = (cos theta)/(sin theta) = -1/0 = "undefined"#

To sum up:

#sin theta = 0, cos theta = -1# #csc theta = "undefined"; sec theta = -1# #tan theta = 0, cot theta = "undefined"#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.