1). How do Trig Graphs relate to the Unit Circle?Trig graphs relate to the unit circle in that they follow the same pattern that is defined by ASTC in the Unit Circle in their periods, which is the length of each cycle in the graphs. As a result of this, the trig graphs are basically just "unraveled" versions of the unit circle.
Why do sine and cosine have periods of 2 pi and tangent and cosine have periods of just pi?
- This is can be explained through the Unit Circle and ASTC patterns that are repeated through the periods, as mentioned above. In order to fully understand this, the picture below shows this relationship.
- On the top is ASTC and the Unit Circle, and on the bottom are the trig graphs. Note how for sine and cosine it takes a full revolution of the unit circle in order to repeat the pattern of the signs. A full revolution is 2 pi. Since trig graphs follow the same pattern, it would take 2pi to repeat the +/- pattern and thus the period would be 2pi. However, take note of the tangent and cotangent pattern. This pattern only requires 1/2 a revolution, or just pi, to repeat itself and thus the period is just 1 pi for tangent and cotangent.
- When we look at the different types of graphs, we notice that sine and cosine have infinite domain but restricted range. This calls need for amplitudes, which define the range of such graphs.
- But why does this occur? If we refer back to the trig ratios of the Unit Circle, we know that sine and cosine are the only two with denominators of r, which is always 1 on the Unit Circle. As a result, Sine and Cosine could be no bigger than 1, otherwise the ratio would cause for a point not on the unit circle resulting in no solution. Thus, amplitudes are needed in trig graphs in order to ensure that the graph stays in this zone.
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