BQ#4 – Unit T Concept 3 Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill?

             

Tangent: Tangent go up because the asymptotes of tangent are found wherever x=0 (tan=y/x) and that is in pi/2 and 3pi/2. Since you can't touch the asymptotes, the only to go through the pattern correctly is to go up in the 1st quadrant since it is positive. (if it were to go down then it would have to go through the asymptote). In quadrants 2-3 they would from positive to negative so the whole period fits in between the asymptotes Pi/2 and 3pi/2. The fourth quadrant is negative so it comes from underneath the 0.

Cotangent: Cotangent goes downward because the asymptotes are found at 0, pi, 2pi. the 1st-2nd quadrants are positive and negative so they can complete a complete period in between the asymptotes 0 and pi.  Same thing applies for in between pi and 2pi with 3rd being positive and 4th being negative.  

***REMINDER*** These are just snapshots, the whole graphs are infinite.

BQ #3: Unit T: Concepts 1-3. Sin and Cosine Vs other trig functions.

Sine and Cosine:

Sine and cosine both are the only graphs that don't have asymptotes. There are just continuous graphs. The only difference with these two is that the Sine graph starts at 0 and the Cosine graph starts at 1.

 Tangent and Cotangent:

Unlike Sine and Cosine, these graphs are not continuous. They have asymptotes since they don't have 'r' as a denominator (tan=y/x and cot=x/y) The difference between these and Csc and sec is that csc and sec are more of the reciprocal of sin and cos.

                                                                                                                                   

Sec and Csc:

Secant is related to cosine in the way that it is it's reciprocal. Since it is the reciprocal, they both touch at the amplitudes, and that is in fact the only place they touch. When Cos is equal to zero though, THIS IS WHERE THE ASYMPTOTE IS!!
They are both the same values in each quadrant.

The Asymptote is located where the sin is 0. (csc=r/y) They both start at 1. 1st and 2nd quadrant are positive for both and and 3rd and 4th are negative. One period is generally 2pi just as sin and cosine.

BQ #5: Unit T: Concepts 1-3: Why do sine and cosine NOT have asymptotes, but the other four trig graphs do?

Sine and Cosines are the only trig functions that don't have any asymptotes. This is because the value of sin and cosine will never be undefined. Both Sine and Cosine are divided by '"r." r=1 and so the Sin (y/r) and Cosine (x/r) are always divide by 1.

 Now the others aren't as lucky. Since Secant (r/x) and Cosecant (r/y) are divided by x or y. Whenever x or y is equal to 0 then it will be undefined and this is the reason for Asymptotes to be formed. Same goes for Tangent (y/x) and        

Cotangent (x/y)                                                       

(Image here)

BQ#2:Unit T: Intro: Trig graphs relate to the Unit Circle?

The trig graph is simply just the unit circle in a continuos line. For example: the Values for sine are +,+,-,- (from quadrant 1-4 in order.) So that means that the graph for sine will have to start at zero and go all the way up to 1 since that is the highest point that sine can go up to. And then all the way down the -1 since it has to go to the negative in order to finish one of the periods. As seen on the photo below























Period- the period for both Sine and Cosine are 2pi since it needs to complete the cycle of negatives and positives. We already went over he pattern for sine, now Cosine is +,-,-,+. (from quadrant 1-4) so we need all four to just complete 1 period.

***NEVER FORGET THESE GRAPHS ARE INFINITE JUST AS THE UNIT CIRCLE!! THESE ARE JUST ONE PERIOD OF THE WHOLE THING.***

Amplitude: The reason that sine and cosine have an Amplitude of one is because they are over r which is equal to one. Sin=y/r or 1/1 since Sin=1 and so does r. Same applies for cosine and x. 

Reflection Unit Q concept 1-5

1. To verify a trig identity simply means to show how one equation can be manipulated and simplified to equal the other equation. You can't touch the right side of the equation because that is already the simplification of the 1st. equation. When you verify it, it justifies whether the answer would be correct or not.

2. The tips that I really found helpful were the factoring out. GCF is another one that is really helpful to me. Also multiplying by the conjugate and separating of the fraction. GCF more since i tend to have been using it the most. And I cant really explain why or how but it is the easiest one in my head to do.

3. My thought process when verifying a trig identity is 1st. are there any identities here that can be changed or replaced to be simpler. Then, can and should i change into sin and cos? see if there is anything that i can do to cancel and get the minimum amount of trig functions. If they are fractions, then try to get a GCF. I might also separate some fractions. Afterwards i will simplify by using algebra.