How do the trig graphs relate to the Unit Circle?
a. Period? Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
Trig graphs relate to a Unit circle in that it contains quadrants from the unit circle and can be created with the use of a computer cable.
Using all students take calculus Sin is positive in the first and second quadrant while sin is negative in the third and fourth quadrant. For cosine the first quadrant is positive, the second is negative and then it repeats itself by being negative then positive in the third and fourth quadrant. Tangent and Cotangent is positive, negative, positive and then negative. Our unit circle is also consistent of having quadrant angles split by 4 sections on the axis. For sine the first quadrant is to pi/2 then to the other half becomes pi. which is half of the unit circle, going into the 270 quadrant angle is 3pi/4 and completing at 360 is 2pi. Here if you look at the picture the quadrants are divided along the line by the quadrants and their values according to the trig function. When you look at the lines going up and down on the line, this is based on our sin value of the quadrants both being positive and then negative in the 3rd and 4th quadrant. We can see this by my lines indicated on the line, after the second one there's a "valley" in which the sin values are negative so they go below. Sin and cosine are 2pi because they don't have a complete pattern like tan and cot. Sin is positive positive then negative negative, so it has to go through a whole go around the unit circle, same thing applies to cosine in which it doesn't complete its pattern. While tan and cot complete its pattern of being positive negative then positive negative.
b) Amplitude?-How does the fact that sin and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about our unit circle?
Sin and cosine have restrictions on their values as -1<x<1 Since we know the unit circle has an r value of one hence our trig values for sin is y/r and x/r belongs to cos. If we get x or y values greater then -1 or less than -1 we get error and it doesn't work, meanwhile the other trig functions you can mix and match the values of the unit circle.