a. Period?
In sine, the first and second
quadrant are positive while the third and fourth quadrant are negative. If we were to unwrap the unit circle and graph it, the 1st and 2nd quadrant would be above the x axis (positive) and the 3rd and 4th quadrant would be below the x axis (negative). This means that for it to have a positive mountain and negative valley, the period must be 2pi because it goes all around the unit circle.Please look at the first picture for a visual. Cosine is positive in the first and fourth quadrants and negative
in the second and third. Cosine also follows the same rules. So, when it is positive, the graph is above the line and when it is negative, it is below the line. Please look at the 2nd picture for a visual. Therefore, it also takes 2pi to have a completed
cycle. However, tangent and cotangent are positive in the first and third and
negative in the second and fourth quadrant. When we graph this, we can see that it only takes 1pi to
complete the cycle. Please see the third picture.
b. Amplitude
Sine and Cosine graphs have amplitudes of 1 because a unit circle has a radius of 1. If we were to unravel a unit circle, and graph it according to the sin and cosine graph, the amplitudes would be 1. This works for only sine and cosine because they have bounded ranges, which means it has an upper limit and lower limit. The rest of the graphs do not have bounded ranges. They have infinity in the range.
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