BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?Emphasize asymptotes in your response.

1. Tangent
With tangent, we know that the unit circle ratio would be sine over cosine. If we would put that into variables, y over x. When we look at how sine and cosine graphs relate to tangent, we notice that if both sine and cosine are positive, then tangent would be positive; if one is negative, then tangent is negative; if both are negative, then tangent is positive. That is basic division rules. Unlike sine and cosine, tangent graphs have asymptotes (which are areas that we cannot touch; undefined). What we need to realize is that the sine over cosine ratio is responsible for those asymptotes: if we divide any number by 0, it would come out as undefined. If cosine (x) were to be equal to 0, that would result in an asymptote. If we were to look at a unit circle, we should know that the x value would equal 0 at 90 degrees and 270 degrees. Converting back to the tangent graph and radians, we know that our asymptotes would be at pi/2 and 3pi/2.

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

2. Cotangent
With cotangent, we face the reciprocal unit circle ratio of cosine over sine, or x over y. When we look at how cosine and sine graphs relate to tangent, we notice that if both cosine and sine are positive, then cotangent would be positive; if one is negative, then cotangent is negative; if both are negative, then cotangent is positive. Unlike cosine and sine, cotangent graphs also have asymptotes Cotangent graphs are identically related to cosine and sine as its reciprocal, but have different asymptotes. If sine (y) were to be equal to 0, that would result in an asymptote. If we were to look at a unit circle, we should know that the y value would equal 0 at 0 degrees and 180 degrees. Converting back to the cotangent graph and radians, we know that our asymptotes would be at 0 and pi.

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

3. Secant
Unlike tangent and cotangent, secant is only related to the cosine graph. The unit circle ratio of secant is 1/cosine. If cosine were to have positive values on its graph, so would secant, and vice versa. Because secant is the reciprocal of cosine, we notice that if we were to have a low value on the cosine graph, the reciprocal of that value would be bigger on the secant graph, and vice versa. We should also notice that each curve of the secant graphs touch the "mountains" and "valleys" of the cosine graph. Looking at a graph, if we were to have a value of 0 on the cosine graph (0/1), we should know that that is where our asymptotes would be for the secant graph(1/0 = undefined).

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

4. Cosecant
Again, unlike tangent and cotangent, cosecant is only related to the sine graph. The unit circle ratio of cosecant is 1/sine. If sine were to have positive values on its graph, so would cosecant, and vice versa. Because cosecant is the reciprocal of sine, we notice that if we were to have a low value on the sine graph, the reciprocal of that value would be bigger on the cosecant graph, and vice versa. We should also notice that each curve of the cosecant graphs touch the "mountains" and "valleys" of the sine graph. Looking at a graph, if we were to have a value of 0 on the sine graph (0/1), we should know that that is where our asymptotes would be for the cosecant graph(1/0 = undefined).

 

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

 

 

BQ #4: Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill? Use unit circle ratios to explain.

If we refer to the unit circle and ASTC, we know that tangent and cotangent are positive in quadrants 1 and 3 and negative in quadrants 2 and 4. If we were to break apart the unit circle into a line, we would get + - + - values for both trig functions. And unlike the other trig functions, tangent and cotangent have a period of pi rather than 2pi.

Tangent has the ratio of sin/cos (y/x), so we get asymptotes whenever cosine is equal to zero. On the unit circle, that would be the values of pi/2 and 3pi/2. If we were to draw out the graph, we notice that it goes "uphill" because each asymptote compresses an area where the graph would start with negative values and into positive values.

Cotangent has the ratio of cos/sin (x/y), so we get asymptotes whenever sine is equal to zero. On the unit circle, that would be the values of 0 and pi. If we were to draw out the graph, we would notice that it goes "downhill" because each asymptote compresses an area where the graph would start with positive values and into negative values.

BQ #5: Unit T Concepts 1-3

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

Sine and cosine do not have asymptotes because their ratios: y/r and x/r. This means that the radius (r) will always be equal 1, and will never have a 0 as the denominator. Sine and cosine will not encounter asymptotes because we will never have an undefined answer (0 divided by any number).

Unlike sine and cosine, the other four trig graphs do include asymptotes. Cosecant has a ratio of r/y. The "y" value can be 0 in :(1,0) or (-1,0). Secant has a ratio of r/x and x can be 0 in : (0,1) and (0, -1). Cotangent and tangent have the ratios of x/y and y/x, which means either numerator can have the value of 0.

A crucial thing to remember is that cotangent and and cosecant have the same denominator in their ratios (y), which means they will have the same asymptotes. Tangent and secant will have the same asymptotes because their denominators in their ratios (x).

BQ #2: Unit T Concept Intro

How do the trig graphs relate to the Unit Circle?

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

When we refer back to the Unit Circle involving sine, cosine, and tangent, we remember ASTC to remember which trig function is positive for each quadrant.

If we refer to ASTC for sine, we know that the function would be positive in quadrants 1 and 2, and negative in quadrants 3 and 4. This creates the repetition of + + - -. When dealing with periods, we have to have a completed pattern. And since it takes the whole unit circle, which equals 2pi, to make that pattern repeat itself, the period is 2pi.

                                  http://www.regentsprep.org/Regents/math/algtrig/ATT5/sine77.gif

This idea also goes for cosine. The function is positive in quadrants 1 and 4, and negative in quadrants 2 and 3. This gives us the repetition of + - - +. Because it does not have a repeating pattern, a cosine period is also 2pi.
 

 

    http://www.regentsprep.org/Regents/math/algtrig/ATT5/unitcirclecowine.gif

Unlike sine and cosine, a tangent period is only pi. Referring to ASTC, tangent is positive in quadrants 1 and 3 and negative in quadrants 2 and 4, which gives us the pattern of + - + -. Comparing to sine and cosine, we already have a pattern, but instead of going all the way around the unit circle for a period, we only need to go half way, or pi.

                                        http://www.clarku.edu/~djoyce/trig/tan.gif

Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig function)relate to what we know about the unit circle?

Amplitudes are half the distance between the highest and the lowest points on the sine and cosine graphs. Sine has the ratio of y/r and cosine has the ratio of x/r. R can only equal 1 (the radius of a unit circle is 1), which means the values x and y can only go up to 1. When dealing with the value of sine and cosine, we should know that sine and cosine have a value range of -1 to 1; anything out of that range would leave the trig function undefined. Now we look at other trig functions: tangent is y/x. We are not restricted to value range of -1 and 1 because we can divide by any other number. Cotangent is the reciprocal (x/y) of tangent. For cosecant and secant, they have the ratios of r/y and r/x. You can divide your "r", which is 1, by a smaller number, and get any value bigger than -1 or 1. That is why those other functions do not have amplitudes, but instead asymptotes.

Reflection #1: Unit Q: Verifying Identities

1.What does it actually mean to verify a trig identity?

To verify a trig function means to have a different version of the same trig function. We could have different trig functions in equations or fractions, but eventually reduces to one trig function.

2.What tips and tricks have you found helpful?

To get through this unit, it is crucial to memorize all the trig identities. It was easy to do because we had already learned and memorized all the reciprocal identities from a past unit. Also remember that reciprocal and ratio identities are able to power up or down, but not Pythagorean identities.

3.Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you.

To get through to verify a trig function, the first thing I do is look for any identities to simplify the function. I would look for any greatest common factors, cancel out, combine like terms, or it dealing with fractions, I would multiply the fraction by the conjugate to get a common denominator. If there are any tan or cot functions, I could use ratio identities to convert them to sin and cos, but one thing I remember is not to square anything because I am only allowed to square both sides.