BQ #3 Unit T Concepts 1-3

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

   Tangent's ratio identity is sin/cos. As cosine equals zero, tan is undefined meaning it will not touch wherever cosine equals zero. As sine equals zero however, tan will intersect the x-axis because tan will equal zero. 

b) Cotangent?

   Cotangent's ratio identity is cos/sin. As sine equals zero, cot is undefined meaning it will not touch wherever sine equals zero. As cosine equals zero however, cot will intersect the x-axis because cot will equal zero. 

c) Secant?

   Secant's reciprocal identity is 1/cos. That means whenever cosine equals 0, then secant will be undefined and will never touch wherever cosine equals 0. When cosine equals 1, secant will equal 1 thus the amplitudes of secant and cosine will touch. 

d) Cosecant?

   Cosecant's reciprocal identity is 1/sin. That means whenever sine equals 0, cosecant will be undefined and will never touch wherever sine equals 0. When sine equals 1, cosecant will equal 1 thus the amplitudes of cosecant and sine will touch. 

B#5 Unit T Concept 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? 
   

   To have asymptotes your trig function has to be undefined at some point which is when the denominator equals 0. 

   Sine and Cosine don't have asymptotes because when you take a right triangle from the unit circle starting at (0,0) sine and cosine have the denominator of r which is the radius of the unit circle. The radius is always 1. Since the radius can never be zero, sine and cosine are never undefined. 

   All other trig functions have asymptotes because their denominators can equal 0 and thus make the trig function undefined. Also with the same variable denominators like tan and csc, they have the same asymptotes! 

BQ#4 Unit T Concept 3

Why is a "normal" tangent graph uphill, but a "normal" tangent graph downhill? 

   If you take a look at the graph of tangent (top), the asymptotes starting from (0,0) are at pi and 3pi/2. These are also where cosine is zero and touches the x-axis. When remembering trig identities, tan = sin/cos. If Cosine is zero, that means tan is undefined and it's graph cannot exist where cosine is zero which is why the asymptotes are there. It goes uphill because starting from (0,0) the first quadrant it 0 to pi/2 where tan is positive. The graph goes up and along the asymptote positively. After the asymptote, tan is negative II quadrant but positive III quadrant so the graph starts negative up against (but not touching!) the asymptote and then curving up positive against the asymptote making an uphill curve. 

   For the cot graph (bottom), the asymptotes are shifted! The asymptotes are now at 0 and pi, where sine is 0. The identity states that cot = cos/sin, the reciprocal of tan's identity. Where cosine is zero, cot is undefined and the cot graph cannot touch where cosine is zero. The graph goes downhill because the asymptote starts at (0,0) and quadrant I starts there. Thus the graph will start against the asymptote going down into quadrant II which is negative and staying there because of the next asymptote at pi. This makes the downhill curve. 

BQ#2 Unit T: Concept 1

How do Trig Graphs relate to the Unit Circle?
a) Period-Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?

   According to the unit circle, sin is positive in quadrants I and II whereas quadrants III and IV are negative making the pattern +, +, -, -. Cos is positive in quadrants I and IV whereas quadrants II and  III are negative making the patter +, -, -, +. It takes a whole unit circle to complete a pattern until it repeats itself. When you look at a unit circle, the circumference is 2pi. If you cut it at one point and laid it flat, it would still be a length of 2pi. That is basically a sine and cosine graph's period. You can also see the patterns represented on the graph. 

   However, according to the unit circle, tan/cot is positive in quadrants I and III whereas quadrants II and IV are negative. The pattern repeats at +, -. This is only halfway of the unit circle which is pi and thus not 2pi like sin and cos. 
b) Amplitude?-How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?
   This is because in the unit circle they are restricted to (0,1), (-1,0), (1,0),  (0,-1) or restricted from -1 to 1 which is the radius of the Unit Circle. This creates walls for the sin/cos graphs and thus they have amplitudes of one. 

Reflection #1: Unit Q Concept 5: Verifying Trig Identities

SP#7: Unit Q Concept 2: Pythagorean/Ratio/Reciprocal Identities with SOH CAH TOA

Please see my SP7, made in collaboration with Trisha M., by visiting their blog here.  Also be sure to check out the other awesome posts on their blog.