Quadrants and Angles (page 2 of 3)

Sections: Introduction, Worked Examples (and Sign Chart), More Examples

The thought process for the exercise above leads to a rule for remembering the signs on the trig ratios in each of the quadrants. In the first quadrant, all the values (x, y, and r) are positive, so All the trig ratios are positive. In the second quadrant, the x-values are negative, so x/r and y/x are negative; only y/r is positive, so only the Sine is positive in QII. In the third quadrant, each of x and y is negative, so x/r and y/r are negative; only y/x is positive, so only the Tangent is positive in QIII. In the fourth quadrant, the y-values are negative, so y/r and y/x are negative; only x/r is positive, so only the Cosine is positive in QIV.


You're probably wondering why I capitalized the trig ratios and the word "All" in the preceding paragraph. I did that to explain this picture:


The letters in the quadrants stand for the initials of the trig ratios which are positive in that quadrant.


axis system, with 'A' in QI, 'S' in QII, 'T' in QIII, and 'C' in QIV

Some people remember the letters using the word "ACTS", but that's the reverse of normal (anti-clockwise) trigonometric order. Others remember the letters with the word "CAST", which is the normal rotational order but doesn't start in the usual (first-quadrant) starting place. To start in the usual spot and rotate in the usual direction, still others use the mnemonic "All Students Take Calculus" (which is a bit ironic when you're in a trig class). Use whichever method works best for you.

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