BQ #4 - Unit T Concepts 1-3: Why is tangent uphill while cotangent downhill?

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

This picture represents the different quadrants and the graph of cosine. We graph cosine because we know based off of the unit circle ratios that tangent is undefined when cosine equals zero (tan=sin/cos). That means there are asymptotes where the graph of cosine hits the x-axis (pi/2, 3pi/2). Due to this, the graph of tangent must be broken between quadrants 1 and 2, it cannot continue because of that asymptote. However, it continues between quadrants 2 and 3, as there is no asymptote at pi (cosine does not touch the x-axis/equal zero). This means it can follow a downhill then uphill shape, giving the normal tangent graph an "uphill" look.

Cotangent is undefined when sin equals zero since the unit circle ratio of cotangent is (cos/sin). This means there are asymptotes where sin=0, a.k.a. where it hits the x-axis (0, pi, and 2pi). Since there is no asymptote between quadrants 1 and 2 (sine does not touch x-axis/does not equal zero), the graph of cotangent can be continuous. The only way we can graph this is by following a downhill pattern, giving cotangent a "downhill" look.

As seen by the images, the graph of tangent and cotangent differ because of the location of the asymptotes. In tangent, the pattern is "broken" by an asymptote, causing it to have an uphill look. Cotangent is not broken by asymptotes and thus ensues a downhill look.

REFERENCE: All graphs from Desmos.com and template from Mrs. Kirch. YOu can view more of her blogs here.

BQ#3 – Unit T Concepts 1-3: How do sin/cos graphs relate to other trigonometric graphs?

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

Tangent:

We know from our trig identities that the ratio for tangent is sin/cos. In the first quadrant, we see that sine and cosine are positive; therefore, tangent must also be positive. 

 The second quadrant features cosine as negative. Based off of tangent's ratio (sin/cos), when cosine is negative tangent must also be negative.

 Both sine and cosine are negative in quadrant three. If the ratio is sin/cos and they are both negative, then the ratio becomes positive. That it why the graphs of sin and cosine are negative while the graph of tangent is positive.

We know that asymptotes result when we have an undefined value. That undefined value comes when we divide by zero. If the ratio for tangent is sine/cosine, we would get undefined when cosine is zero. Based off our graph, cosine is zero at pi/2 and 3pi/2. We can tell based off of tangent's graph that it does limit itself to those asymptotes. Our plotted points show the locations where cosine is equal to zero.

Cotangent:

 If we look at quadrant one, both sine and cosine are positive. The ratio for cotangent is cos/sin. If they are both positive, then the graph of cotangent must also be positive. Our graph proves that that is true, cotangent is positive in quadrant one.

Cosine is negative in quadrant two. If the graph of cotangent is a ratio that contains cosine (cos/sin), then the graph for cotangent must be negative also if one of its component is negative.

The graphs for sine and cosine are both negative in quadrant three. If a ratio has negative in both numerator and denominator, then the ratio becomes positive. Because both sine and cosine are negative and the ratio for cotangent is (cos/sin), the graph for cotangent is positive.

In the last quadrant sine is negative. That makes the graph for cotangent negative since it would make the ratio for cotangent (cos/sin) negative.

If the ratio for cotangent is cos/sin, we would get undefined when sine is equal to zero. Whenever we get undefined we have an asymptote, so we should see the graph limit itself at those points. The plotted points mark where sine is equal to zero (0, pi, and 2pi). Those points are where sine is zero (where the graph of sine touches the x axis).

Secant:

The ratio for secant is the reciprocal of cosine. That means it is 1/cos. If cosine is positive, then secant must also be positive. Since it is a reciprocal, the graph goes up really high (we are dealing with small fractions).

Because cosine is negative, the reciprocal also becomes negative. This causes the graph of secant to be negative.

Cosine is also negative in quadrant three, causing the graph of secant to be negative again. Since there is no asymptote between quadrant two and three (the graph of cosine does not touch the x-axis), we are able to continue the graph of secant from the last quadrant.

Cosine is positive in quadrant four, making its reciprocal also positive.

If secant is the reciprocal of cosine, it will have asymptotes wherever cosine equals zero. Cosine is zero when its graph touches the x-axis. We see that the symptotes are at pi/2 and 2pi. The graph limits itself as it nears these asymptotes.

Cosecant:

The first quadrant presents cosecant as positive. The ratio for cosecant is 1/sin (the reciprocal of sine). If sine is positive, csc's graph will also be positive.

Sine is positive in quadrant two also, making its reciprocal positive. Since there is no asymptote in quadrants one and two (sine does not touch the x-axis), the graph of the parabola continues.

Sine is negative in quadrant three, causing its reciprocal to be negative. 

Quadrant four features sine as negative, making the reciprocal also negative. Since there is no asymptote in quadrants three and four (sine does not touch the x-axis in between),. the parabola continues.

Cosecant is the reciprocal of sine. That means it will have undefined values whenever sine is zero. Sine is zero at the places where it touches the x-axis, which are outlined as plotted points in this graph. That means the graph will limit itself around these points.

REFERENCE: All graph images obtained from Desmos.com and template from Mrs. Kirch. See more of her posts here.

BQ#5 – Unit T Concepts 1-3: Graphing trig functions

Why no asymptotes for sine/cosine?

The trig functions that we have come to know off the unit circle can also be expressed through ratio identities. These identities allow us to make sense out of multiple trigonometric functions. We know that when a ratio has a denominator of zero, that value is undefined. This undefined results in asymptotes when graphing trig functions.

The ratios for sine and cosine are opposite/hypotenuse (y/r) and adjacent/hypotenuse (x/r) respectively. If we were trying to find asymptotes for those two graphs, we might try to find an instance where r equals zero. However, r is not zero when observing a unit circle because the radius is always one. 

If r can not be zero, then sine and cosine can never be undefined. Because they can not be undefined, sine and cosine do not have asymptotes.

BQ#2 – Unit T Concept Intro

How do trig graphs relate to the Unit Circle?

The period for sine and cosine is 2pi whereas the period for tangent and cotangent is pi because that's the distance it takes for the patterns of sine and cosine to repeat. It takes them a whole revolution of the unit circle (2pi) to repeat their patterns of ++-- for sine and +--+ for cosine. In tangent and cotangent it only takes half a revolution of the unit circle (pi) to repeat the patterns of +- for tangent and +- for cotangent. The rest of the revolution is not counted as a period because it is the same as the first pair.

Sine and cosine have amplitudes of one because that is as far as the unit circle extends. The unit circle has a radius of one and therefore any value of sine and cosine must be between the parameters of -1 or 1. This means that is the highest/lowest values cosine and sine can have since their ratios are both over r. Other trig functions don't have amplitudes because they don't have restrictions. Meaning theta of any of the other trig functions can be equal to any numerical value due to their ratios that are not over r. 

Reflection# 1: Unit Q: Verifying Trig Identities

Verifying Trigonometric Identities

1. To verify a trigonometric identity means to simplify an expression until it looks like what they want you to verify it to. This is not the same as simplifying because in verification, you already know what you want your expression to look like; you are simply doing everything you see fit in order to make the expression match it. Verification often involves simplifying an expression so that it equals a simpler expression or it could also verify to the number 1 and many other variations.

2. The tips and tricks I have found helpful are manipulating expressions so that I can find pythagorean identities. When I identify a pythagorean identity, my problem usually goes in the right direction. When I get something that looks similar to a pythagorean identity, I take out a one and the result is usually something found on my Unit Q SSS cover sheet. When a problem involves different trigonometric functions, I like to convert everything to sine and cosine because usually something cancels out.

3. My thought process and steps I take in verifying a identity involve splitting the fraction if it is a polynomial numerator with a binomial denominator. I also like to get everything to equal 0 when I am working on concept 2 and 4. I also like to look at my verification's trigonometric function so that I know what I am trying to have in the end. This helps me convert everything to the appropriate function and also which things I need to cancel out.