cast rule behavior

Question

The CAST Rule describes the sign of the three primary trigonometric ratios as the angle varies from 0° to 360°. Describe how the CAST Rule relates to the behaviour of the graphs of each of y = sin(x), y = cos(x) and y = tan(x).

Answer 1

360 degrees is divided into 4 quadrants: Q1: 0-90; Q2=90-180: Q3=180-270; Q4=270-360.

In Q1, all trig functions are positive; in Q2, only sine is positive (sin(X)=sin(180-X));

in Q3, only tangent is positive and in Q4 only cosine is positive.

CAST=cosine,all,sine,tangent=Q4,Q1,Q2,Q3.

Graphically cosine and sine look similar but they are displaced by a phase difference of 90 degrees.

Tangent resembles neither cosine nor sine, but is nevertheless periodic in that the pattern repeats.

The red curve is y=sin(x), blue is y=cos)x), green is y=tan(x) (asymptotes are shown as green vertical lines).

Between the y-axis and the first green line to the right all functions are positive (Q1), where the red and green curves intersect further to the right, we have Q2, Q3 is where the green curve is positive, up to the next vertical green line, and Q4 goes off the picture, but Q4 is also up to the first green vertical line to the left of the y-axis, followed further left by Q3, etc. The regular pattern continues indefinitely repeating every 360 degrees. The gap between the green verticals is 180 degrees.