> I am having some difficulty with the following ACT > practice problem: > > For X, an angle whose measure is between 270° and > 360°, cosX=5/13. Which of the following equals tanX? > A. -5/12 > B. -5/13 > C. 5/13 > D. 5/12 > E. 12/13 > > The book I am using provides the answer and an > explanation, but the explanation uses process of > elimination to find the answer without actually > describing a method of solving the problem. I would > like to understand how to actually find the answer. > Can someone lend a helping hand? > > Thanks!
I don't know what ACT is, but this is absolutely elementary trig, which you must understand if you want to do anything at all in trig.
Draw coord axes yox and a circle of radius r (always +ve)centred at the origin. Angles are measured anticlockwise from ox. The quadrants are called 1, 2, 3, 4 (or I, II, III, IV). For an arbitrary position of the radius vector, project it onto the x-axis to give a segment X long. The join from the outer end of X to the outer end of r is Y long. The sign of X and Y come from the coord axes. In quadrant 1 all trig functions are +ve. In quadrant 2 sin is +ve. In quadrant 3 tan is +ve. In quadrant 4 cos is +ve.
For the example you give with 270 < theta < 360, r lies in the 4th quadrant, and r=+13, X=+5 since cos(theta)=5/13 By pythagoras Y = -12 .: tan(theta) = Y/X = -12/5 so the answer is none of those given!
Also cos(theta)=5/12 .: theta = + or - 67.38deg or 292.62deg
tan(67.38) = 12/5 = 2.4
tan(-67.38) = -12/5 = -2.4
tan(292.62) = -12/5 = -2.4
I hope this helps. Please feel free to ask more questions if you need more help.