BQ#2: Unit T Concept Intro

1. Trig graphs relate to the unit circle because if we were to unwrap the circle, it would still have quadrant 1,2,3, and 4. This "unwraping" would be as if it were in a graph. For example, sine is positive in the first and second quadrant of the unit circle, so in the graph that section would be above the x-axis. These values would be positive from 0 to pi and the values would be negative until it would get to 2pi because quadrant 3 and 4 in the unit circle are negative. This cycle goes on forever, this is just a portion of it. For cosine, it is pretty much the same idea. Cosine is positive in the 1st quadrant, negative in the 2nd and 3rd quadrant, and positive in the fourth quadrant. If the unit circle were to be unwrapped once again, the line would be above the x-axis until it reached 90 degrees, or pi/2, and then it would be below the x- axis until 270 degrees, or 3pi/2, was reached and finally, it would be positive from 3pi/2 to 0. This would be one complete period. The period of sine and cosine is 2pi because it will take the entire distance of the unit circle to reach one complete period whereas tangent and cotangent only take half of that, which is just pi to complete a period. To continue further explanation, tangent is positive in the 1st quadrant, negative in the 2nd quadrant, positive in the 3rd quadrant, and negative in the 4th quadrant. This means that from 0 to pi/2, the graph is positive, from 0 pi/2 to pi, the graph is negative, from pi to 3pi/2, the graph is positive, and from 3pi to 0, the graph is negative.
(pictures below for further explanation)

Sine

 

Cosine

 

 

Tangent

 

 

2. Amplitudes are half the distance between the highest and lowest points on the graph. Sine and Cosine have amplitudes because the trig ratio for sine is y/r and for cosine it is x/r, and we know that r will always be 1. The center of the unit circle is (0,0) and its radius is one, therefore, this is how much sine and cosine can extend to. This is where the 1 and -1 come from; these are the only numbers that work, anything that is outside these numbers will not funcion. The rest of the four trig function's ratio is not over 1, so they are not as restricted. Because these values are not restricted and go on forever, they cannot have an amplitude. They will always be from negative infinity to positive infinity, so this does not allow them to have a highest and lowest point.