I'm given Cos(theta) = -2/3 and Tan(theta)=1/2 *sqrt(5) and I'm asked to find sin, sec, csc, cot,

Think of a right triangle, draw it. You have the long side (Hypotenuse) and you have the short sides (the legs). Put a Theta symbol in one of the two non-90degree angles. The short side across from your theta symbol we will call O (O for opposite). The short side (not the hypotenuse) that is next to your theta symbol we will call A (A for adjacent). The hypotenuse we will denote as H. The following trig functions can be defined as follows: sin(theta)=O/H csc(theta)=1/sin(theta) cos(theta)=A/H sec(theta)=1/cos(theta) tan(theta)=O/A cot(theta)=1/tan(theta) If cos(theta) = -2/3, and cos(theta)=A/H, then you know that A=-2 and H=3. Use the Pythagorean theorem (a^2+b^2=c^2), in this case, (A^2+O^2=H^2) to find the last side. Once you know the last side, you can finish defining the last remaining trig functions.

That's very helpful. However the way the question is given it doesn't allow for such simple calculations. It asks for the quadrant which I found to be 3 then using your method I've been marked incorrect. Its the tan(theta)= 1/2 * sqrt(5) that is throwing me off.

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