PreCalc 2012 Student Blog: Unit T Big Question Blog Post #1

1) 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?

Well for sine and cosine, they have a period of 2pi because for both, their pattern of being positive or negative doesn't repeat until after four marks. On the unit circle, our quadrant angles (0,90,180,270, 360) in radians are 0, pi/2, pi, 3pi/2, and 2pi. Sine is positive in quadrant 1 and quadrant 2, thus meaning that it is negative in quadrant 3 and 4 which makes the pattern + + - -. Cosine is positive in quadrant 1 and quadrant 4, thus meaning that it is negative in quadrant 2 and 3 which makes thee pattern + - - +. As you can see, on the unit circle, the pattern for these two trig functions repeats only after one full revolution and since one revolution around the unit circle is 360 degrees, it has a period of 2pi. The same is on the graph except that the graph shows the unit circle in a different form on the x-axis. One period on a graph is like saying one revolution on the unit circle.

Source: http://www.upv.es/~rfuster/xpicture/functiongraphs.html

However, tangent and cotangent have a period of pi because their pattern of being positive or negative only repeats after two marks, or half of a revolution. Tangent is positive in quadrant 1 and 3 and is negative in quadrant 2 and 4. Cotangent is the same. This means that the pattern is + - + -. According to the graph, you can notice after every two marks, tangent repeats.

Source: http://www.analyzemath.com/trigonometry/properties.html

b) Amplitude?-How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the unit circle?

First of all, sine and cosine have amplitudes of one because that is where the minimum and maximum points are, meaning that unlike the domain on a sine/cosine graph, the range is specific. The domain is all real numbers.Unlike sine and cosine, tangent and the rest of the trig functions have asymptotes, so when the functions are graphed in between the asymptotes, the vertex indicates where the function begins to go up to infinity or down to negative infinity.