BQ#4: Unit T Concept 1-3

From the Unit Circle, we know that tangent and cotangent are both positive in the first and third quadrants and negative in the second and fourth quadrants. Tangent=sine/cosine while cotangent =cosine/sine. Tangent would have asymptotes wherever cosine equals zero (whenever it is undefined) In contrast, cotangent would have asymptotes wherever sine equals zero. The graphs of tangent and cotangent are different because of their different asymptotes. Even though the asymptotes are different, they both still have to be positive in the first and third quadrant and negative in the second and fourth quadrant. Tangent is going uphill and cotangent is going downhill in order to not touch the asymptotes and also to follow the positive/negative rules of the quadrants in the Unit Circle.

(Red=Quadrant 1, Green=Quadrant 2, Orange= Quadrant 3, Blue= Quadrant 4)

 

Tangent

 

Cotangent

 

BQ#3: Unit T Concepts 1-3

Tangent

Sine and cosine relate to tangent because when we look back at the Unit Circle, sine and cosine are positive in quadrant one. Based on our knowlwdge of identities, we know that tangent=sine/cosine and since sine and cosine are both positive in the first quadrant, then a positive divided by a positive will result in a positive, which is what tangent is in this quadrant as well. This will cause the tangent graph to be above the x-axis in the first quadrant. In the second quadrant, sine is positive and cosine is negative. If tangent=sin/cos, then this means that tangent will be negative in the second quadrant, since a positive divided by a negative will result in a negative. In the picture, we see that the tangent graph is below the x-axis. In the third quadrant, both sine and cosine are negative, so when you divide two negatives, you get a positive, and tangent is positive as a result of that. In the picture, we see that tangent is once again above the x-axis. In quadrant four, sine is negative and cosine is positive. When divided, this gives you a negative, so tangent will be negative and this will cause it to be below the x-axis. Also, because tangent is sine/cosine, tangent will have asymptotes wherever cosine equals zero on the graph.
(Red=Quadrant 1, Green=Quadrant 2, Orange= Quadrant 3, Blue= Quadrant 4)

Cotangent

In quadrant one, sine and cosine are both positive, so because cotangent=cosine/sine, when this is divided, cotangent will be positive, or above the x-axis. In quadrant two, sine is positive and cosine is negative, so a negative divided by a positive gives you a negative, so cotangent is negative in this quadrant, or below the x-axis. In quadrant three, cotangent will be positive because sine and cosine are bothe negative, so when they are divided, you get a positive. Cotangent will be above the x-axis here. In quadrant four, cosine is positive and sine is negative, so when they are divided, cotangent will result in being negative, or below the x-axis. Also, Cotangent will have asymptotes wherever sine equals zero. Asymptotes only occur when we have undefined, or whenever we divide by zero, so whenever sine=0, there will be an asymptote.

 

 

Secant





In quadrant one,cosine is positive, so secant is positive as well. Secant is the reciprocal of cosine. The cosine graph has very small plotted points, so the reciprocal of those numbers is very large. This is why secant goes up so high. The points plotted in the secant graph will be the reciprocals of the cosine graph. Secant will have asymptotes wherever cosine equals zero. In quadrant two, cosine is negative, and so is secant, so secant will be below the x-axis.  Therefore, secant also has a point here as well. In quadrant three, cosine is negative, so secant is negative too, so secant will be below the x-axis. In quadrant four, cosine is positive, as well as secant, so secant will be above the x-axis.

 

Cosecant

In quadrant one and two, sine is positive, so cosecant is positive as well. The cosecant graph will touch the mountain of the sine graph and as it goes higher, it gets closer to the asymptotes. In quadrant three and four, sine is negative, so cosecant will be negative as well, or below the x-axis. The cosecant graph is shaped where the asymptotes are and the asymptotes are where the sine graph is. Cosecant is one over sine, so whenever sine equals zero, cosecant will be divided by zero. Dividing by zero means getting undefined, and undefined means that there will be an asymptote.





BQ#5: Unit T Concepts 1-3

Sine and cosine graphs do not have asymptotes because their denominator is 1, going back to Unit Circle ratios.
Sin=y/r
Cos=x/r
Csc=r/y
Sec=r/x
Tan=y/x
Cot=x/y
As we see above, the only ratios that have r as the denominator are sine and cosine. R means that the numerator will always have 1 as the denominator, unlike all of the other trig functions. The other trig functions can be either greater or less than 1, so they have asyptotes. Because sine and cosine have the denominator as being 1, they will always be a real number. The other trig functions, however, can end up with 0 as their denominator, causing these functions to be undefined. The result of these undefined functions are what cause these trig functions to have asymptotes

BQ#2: Unit T Concept Intro

1. Trig graphs relate to the unit circle because if we were to unwrap the circle, it would still have quadrant 1,2,3, and 4. This "unwraping" would be as if it were in a graph. For example, sine is positive in the first and second quadrant of the unit circle, so in the graph that section would be above the x-axis. These values would be positive from 0 to pi and the values would be negative until it would get to 2pi because quadrant 3 and 4 in the unit circle are negative. This cycle goes on forever, this is just a portion of it. For cosine, it is pretty much the same idea. Cosine is positive in the 1st quadrant, negative in the 2nd and 3rd quadrant, and positive in the fourth quadrant. If the unit circle were to be unwrapped once again, the line would be above the x-axis until it reached 90 degrees, or pi/2, and then it would be below the x- axis until 270 degrees, or 3pi/2, was reached and finally, it would be positive from 3pi/2 to 0. This would be one complete period. The period of sine and cosine is 2pi because it will take the entire distance of the unit circle to reach one complete period whereas tangent and cotangent only take half of that, which is just pi to complete a period. To continue further explanation, tangent is positive in the 1st quadrant, negative in the 2nd quadrant, positive in the 3rd quadrant, and negative in the 4th quadrant. This means that from 0 to pi/2, the graph is positive, from 0 pi/2 to pi, the graph is negative, from pi to 3pi/2, the graph is positive, and from 3pi to 0, the graph is negative.
(pictures below for further explanation)

Sine

 

Cosine

 

 

Tangent

 

 

2. Amplitudes are half the distance between the highest and lowest points on the graph. Sine and Cosine have amplitudes because the trig ratio for sine is y/r and for cosine it is x/r, and we know that r will always be 1. The center of the unit circle is (0,0) and its radius is one, therefore, this is how much sine and cosine can extend to. This is where the 1 and -1 come from; these are the only numbers that work, anything that is outside these numbers will not funcion. The rest of the four trig function's ratio is not over 1, so they are not as restricted. Because these values are not restricted and go on forever, they cannot have an amplitude. They will always be from negative infinity to positive infinity, so this does not allow them to have a highest and lowest point.

Unit Q: Verifying Trigonometry Reflection

1. Verifying means to solve a problem. To verify something, you must check to so see if it is correct. It meant to solve and go to several steps in order to achieve the correct answer. Verifying means checking different identities and solving them to see if you end up with the correct answer in both sides.
2. Tips would be to turn everything into sin/cos to make everything easier. Finding the common denominator also. Multiplying the conjugate would also make it easier, and that is the denominator binomial. Factoring and separating fractions are also tips to make the problem easier.
3. I look at the problem to see if I can turn anything into sin/cos. sometimes when a problem looks hard, it may already be an identity that equals 1, so that makes it a lot simpler. Memorizing the identities really helps with this because this way you know what it equals. You keep substituting what you know in order to get your final answer. The answer will either have to be as simple as it can be, or equal to the problem you are verifying.