BQ #2: Unit T Concept Intro

How do the trig graphs relate to the Unit Circle?

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

When we refer back to the Unit Circle involving sine, cosine, and tangent, we remember ASTC to remember which trig function is positive for each quadrant.

If we refer to ASTC for sine, we know that the function would be positive in quadrants 1 and 2, and negative in quadrants 3 and 4. This creates the repetition of + + - -. When dealing with periods, we have to have a completed pattern. And since it takes the whole unit circle, which equals 2pi, to make that pattern repeat itself, the period is 2pi.

                                  http://www.regentsprep.org/Regents/math/algtrig/ATT5/sine77.gif

This idea also goes for cosine. The function is positive in quadrants 1 and 4, and negative in quadrants 2 and 3. This gives us the repetition of + - - +. Because it does not have a repeating pattern, a cosine period is also 2pi.
 

 

    http://www.regentsprep.org/Regents/math/algtrig/ATT5/unitcirclecowine.gif

Unlike sine and cosine, a tangent period is only pi. Referring to ASTC, tangent is positive in quadrants 1 and 3 and negative in quadrants 2 and 4, which gives us the pattern of + - + -. Comparing to sine and cosine, we already have a pattern, but instead of going all the way around the unit circle for a period, we only need to go half way, or pi.

                                        http://www.clarku.edu/~djoyce/trig/tan.gif

Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig function)relate to what we know about the unit circle?

Amplitudes are half the distance between the highest and the lowest points on the sine and cosine graphs. Sine has the ratio of y/r and cosine has the ratio of x/r. R can only equal 1 (the radius of a unit circle is 1), which means the values x and y can only go up to 1. When dealing with the value of sine and cosine, we should know that sine and cosine have a value range of -1 to 1; anything out of that range would leave the trig function undefined. Now we look at other trig functions: tangent is y/x. We are not restricted to value range of -1 and 1 because we can divide by any other number. Cotangent is the reciprocal (x/y) of tangent. For cosecant and secant, they have the ratios of r/y and r/x. You can divide your "r", which is 1, by a smaller number, and get any value bigger than -1 or 1. That is why those other functions do not have amplitudes, but instead asymptotes.