Recall that #sintheta = "opposite"/"hypotenuse"#
Hence, the side opposite #theta# in our question measures #1# unit and the hypotenuse measures #4# units.
Since we're dealing with right triangles, we can find the side adjacent #theta# using pythagorean theorem.
Let the adjacent side be #a#.
#a^2 + 1^2 = 4^2#
#a^2 + 1 = 16#
#a^2 = 15#
#a = sqrt(15)#
Now, let's define secant and tangent.
#sectheta = 1/(costheta) = 1/("adjacent"/"hypotenuse") = "hypotenuse"/"adjacent"#
#tantheta = sintheta/costheta = ("opposite"/"hypotenuse")/("adjacent"/"hypotenuse") = "opposite"/"adjacent"#
Applying these definitions:
#sectheta = 4/sqrt(15) = (4sqrt(15))/15#
#tantheta = 1/sqrt(15) = sqrt(15)/15#
The last thing left to do is to find the signs of these ratios. We know that we're in quadrant #II#, where sine is positive, and all the other ratios are negative. Since secant is related to cosine, it will be negative.
So, our final ratios are:
#sectheta = -(4sqrt(15))/15#
#tantheta = -sqrt(15)/15#
Hopefully this helps!
