Unit T-Big Question #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?

Trig graphs are clycical, meaning that they repeat themselves over and over gain. One time through this cycle is called a period. Looking at the Unit Circle we can divide into four quadrants in which each particular quadrant has certain trig functions that are positive or negative depending n the quadrant. We represent this by ASTC, like the picture below.:
                                 

-Sine: Looking at sine we see that sine is positive in the first quadrant, positive in the second quadrant, negative in the third, and negative in the third. In result, the pattern for sine is +, +, -, -. so we see that for the pattern to repeat itself again we have to go through one whole cycle around the unit circle thus 2pi since the untie circle is 2pi all the way around. The following picture shows how the pattern looks like on the unit circle:

-Cosine: Looking at cosine we see that it is positive in the first quadrant, negative in the second, negative in the third, and positive in the fourth. The pattern is +,-, -, +. Like sine we see that it takes one whole cycle around the unit circle for the pattern of cosine to repeat itself so that is why the period is 2pi. The following picture shows how the pattern looks like on the unit circle:

            *Remember a period is when the pattern starts and end. Because it takes both sine and cosine one whole cycle around the unit circle to repeat themselves the period is 2pi. 

-Tangent/Cotangent: Looking at cotangent and tangent both are positive in the first quadrant, negative in the second, positive in the third, and negative in the fourth. The pattern for cotangent and tangent is +, -, +, -. We can see that it takes half the time of sine and cosine for the pattern of tangent/cotangent to repeat itself. We have +, -, and then again +, -., so the pattern begins to repeat itself half way through the unit circle which is pi. The following picture shows how the pattern looks like on the unit circle:

b.) Amplitude? How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?

 --Looking at the Unit Circle we see that the greatest distance we have is one. This puts a limit at to how far the sine and cosine graphs can reach thus making it its amplitude.  The sine of 90 degrees is 1 and the sine of 270 degrees is -1, which further explains that the amplitude is 1. The cosine of 0 degrees is 1 and the cosine of 180 degrees is -1 which again makes the amplitude 1.


citations for images:

http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceTriangles.htm

http://etc.usf.edu/clipart/galleries/617-unit-circles