BQ#2: Unit T Intro - Graphing Sine and Cosine

How do trig graphs relate to the Unit Circle?

• Trig graphs relate to the Unit Circle, because, if unwrapped, it will be a horizontal line with the four quadrants all laid out in a row.

Period? - Why's the period for sine & cosine 2π, whereas the period for tangent/cotangent is π?

• The period for sine and cosine is 2π because when looking at sine, it can be positive in quadrants one and two and negative in quadrant three and four. So, if we lay that circle in a straight line it will look like the picture below. The part shaded red is the first quadrant, green is second, orange is third, and blue is fourth. Quadrants one and two are positive so it's shaped as a "hill" and quadrants three and four are negative so it's a "valley". A period is how long the cycle takes to repeat itself, which in this case is 2π. The same goes for cosine because it's positive in quadrants one and four and negative in two and three, you can clearly see it drawn out in the picture below. The two quadrants in the middle are negative and the sides are positive. As you can see, it only shows half of the curve in quadrant 1 and 2, so together it will be a "hill".
  
  

  

  Sine Graph

  

  Cosine Graph

  ●  The period for tangent and cotangent is π. Let's look at tangent; it's positive in quadrants 1 and 3 and negative in 2 and 4. On a line it will be positive, negative, positive, negative (picture below). The cycle repeats faster because it's positive then negative. It's period is at 180º which is π.

Tangent Graph

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

• Amplitudes are half the distance between the highest and lowest points on the graph. Sine and cosine have amplitudes of one relate to the Unit Circle because they have to be between 1 and -1. This means that the highest and lowest points for either sine or cosine are between 1 and -1; it can't be anything greater. The other trig functions, such as tangent, don't have amplitudes of one because it's ratio is sin/cos. Meaning they don't have a low and high point (you can refer back to the picture above).

Work Cited:

https://www.desmos.com/calculator/hjts26gwst