BQ#3 – Unit T Concepts 1-3

BQ #3

1). How do the graphs of sine and cosine relate to the others?

Tangent

Tangent equals sine/cos as a ratio identity so if one of the graphs is negative then tangent will be negative. If both of the graphs are either positive or negative then the tangent graph will be positive.

Cotangent

The relation is the same as tangent, except the ratio identity is the inverse of tangent, so the ratio identity would actually be cosine/sine.

Secant

It is related to a cosine graph because the reciprocal of cosine is secant.The reason asymptotes are there is once again because of a trig ratio, this time being that sec = 1/cos, and like before, since cosine is the denominator, any place where the cosine value is zero we will have an asymptote.

Cosecant

Just like secant is the reciprocal of cosine, so to are cosecant and sine related in that way. Cosecant is directly related to sine because the reciprical of cosecant is 1/sine.Whenever sine is equal to 0 on the graph, then you will find an asymptote for cosecant there.

BQ #4: Unit T Concepts 3: Tangent and Cotangent Graphs

BQ #4

1) Why is a "normal" tangent graph uphill but a "normal" cotangent graph is downhill?

They have different ratios, and that is why they go uphill and downhill. For example, the ratio for tangent is sine/cosine, this also means that whenever cosine is 0 on a graph that's where tangent will have an asymptote. Cotangent, has the inverse of that ratio, it's cosine/sine and that means that it will have an asymptote when cosine is equal to 0.The difference in the asymptote location causes for the differences in shape. Since cotangent and tangent follow the same ASTC pattern, they both are positive is quadrants I and III and negative in II and IV. In order for cotangent to follow this pattern, it must be downhill.

BQ #5: Unit T Concepts 1-3: Asymptotes

Big Question Blog Post

1. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain

A: An asymptote isA line whose distance to a given curve tends to zero. If we look back at the unit circle we'll see that the trig ratios for sine and cosine are over r, yet r will always be constant, that constant being r=1. Therefore we'll never be able to get asymptotes with sine and cosine. Sine and cosine are never undefined and so they never approach infinity....thus vertical asymptotes for them do not exist. The other trig functions do have asymptotes because they don't have a constant value for the denominator.

BQ #2: Unit T Concept 1 Intro: Trig Graphs and the Unit Circle

Big Question Blog Post: Unit T

1. How do the trig graphs relate to the Unit Circle?

1) The period of a trig function is the horizontal length of one complete cycle. The trig functions of sine and cosine have a period of 2π because their waves repeats every 2π units.  The same can be said of periods of tangeant and cotangeant because their waves repeat at every pi and so that is their period. The following picture shows this. 

Picture illustrates period

2) The amplitude of a sinusoidal function is one-half of the positive difference between the maximum and minimum values of a function. It is the peak from the center of  the graph. At first, trig functions were just related to triangles but now we can relate them to amplitudes and unit circles.  The curves go one unit above and below their midlines (here, the x-axis). This value of "1" is called the "amplitude". It is also illustrated in the above graph

References: http://www.purplemath.com/modules/grphtrig.htm 

Reflection: Unit Q

Reflection

1. To verify a trig function is to make sure that both sides are equal to each other, the same thing, and that it is true, for example 1=1 is true while 2=1 is not true, but instead of simple numbers it's trig identities.

2. One extremely helpful hint was finding out that there isn't just one way to find something. So you and your partner on the test don't have to use the same method to find a problem, but can still get the same answer. It is a good way to check your work.

3.First I would try to convert everything or see if any trig identities can be substituted then I'll try to make the statements equal.

SP7