BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?

1. Tangent

   Tangent has a ratio of (sin/cos). When the tangent graph is labeled based on the quadrants of the Unit Circle,  the positive and negative sign of tangent is determined by the positive and negative sign of sine and cosine. If both sine and cosine is positive on the same quadrant, then tangent will be positive, ending up with an uphill on the graph. If sine is positive and cosine is negative on the same quadrant, then tangent will be negative, ending up with the downhill on the graph. If cosine is negative and sine is positive, then tangent will be negative, which will still have a downhill on the specific quadrant of the graph. If both sine and cosine is negative, then tangent will be positive, which will have an uphill on that specific quadrant of the graph. Tangent has asymptotes when cosine equals to 0. Cosine equals to 0 at pi/2 and 3pi/2, which is where the asymptotes are located and goes on continously at the graph.

2. Cotangent

   Cotangent has a ratio of (cos/sin). Similarly, the cotangent graph is based on positive and negative sign of cosine and sine. However, the difference between tangent and cotangent is that it will have a downhill shape in the beginning of its period rather than an uphill shape like tangent. The reason for its downhill shape is because of where the asymptotes are located. Cotangent has asymptotes when sine equals to 0. Sine equals to 0 at 0 and pi on the Unit Circle. Thus, this is where the asymptotes will be located for cotangent.

3. Secant

   Secant is the reciprocal of cosine and has a ratio of (1/cos). Because cosine is the denominator and can equal to 0, then secant will have asymptotes. Secant has asymptotes at pi/2 and 3pi/2 and so forth. If secant is positive, then the graph will go uphill. If secant is negative, then the graph will go downhill. In other words, if cosine is positive or negative on the graph, so will the secant.

4. Cosecant

   Cosecant is the reciprocal of sine and has a ratio of (1/sin). Because sine is the denominator and can equal to 0, then cosecant will have asymptotes. Cosecant has asymptotes at 0 and pi and so forth. If sine is positive, so will the shape of cosecant of the graph. In other words, if cosecant is positive or negative on the graph, sine will also have to be postive or negative respectively.