Unit T Intro + Concepts 1-3

BQ #2:
Trig graphs relate to the Unit Circle because, if we unwrap the Unit Circle, we get our x-axis on which to plot the points and label the segments. The amount of space that the first quadrant takes up ion the Unit Circle is the amount of space it takes up on a straight line as the x-axis. The quadrant angles (90, 180, 270, and 360) are our distances and measurements. When graphing, though, we use radians instead of degrees because it is simpler on a graphing calculator, but the idea is still the same.

a) Period?: The period for sine and cosine is 2pi whereas the period for tangent and cotangent is pi because that's how long it takes to repeat the pattern. On a Unit Circle, sine is positive in the first and second quadrants and negative in the third and fourth quadrants. When the circle is stretched out in a line, that means that, when graphed, the sine would be positive for the first two segments (first two quadrants) and negative in the last two segments )last two quadrants). The pattern for sin is (+)(+)(-)(-), which extends to the whole Unit Circle to complete a pattern, which measures to 2pi.

The same goes for cosine, which is positive in the first and fourth quadrants. Therefore, on the x-axis, cosine would be positive on the first and fourth segments and be negative on the second and third segments. The pattern for cosine is (+)(-)(-)(+) which lasts a whole Unit Circle measurement and amounts to 2pi as well.

The pattern for tangent and cotangent is (+)(-)(+)(-) because it is positive in the first and third quadrants. Therefore, the pattern is shown to repeat itself already just within one revolution of the Unit Circle: (+)(-) and then (+)(-) again. One pattern starts in the first quadrant and repeats itself starting on the third quadrant. Because the pattern starts and completes itself halfway through the Unit Circle, the period for tangent and cotangent is just pi.

b) Amplitude?: Sine and cosine are the only trig functions with amplitudes because they can't have asymptotes. They will always have amplitudes of one because of the trig ratios for each. Sine as a trig ratio is y/r and cosine as a trig ratio is x/r. "r" is always 1. The restrictions for each can't go higher than +1 and can't go lower than -1.


BQ #5:
Trig graphs only get asymptotes when it is undefined, and a trig function is undefined only when divided by 0. Sine and cosine don't have asymptotes because they can't be divided by 0 and therefore can't be undefined. This is because the trig ratios for sin and cosine have "r" as the denominator: x/r (cos) and y/r (sin). "r" always equals 1 so "x" or "y" for sine and cosine can't be divided by 0 because they're always divided by 1 whereas the rest of the trig functions can be divided by 0 when either "x" or "y" equals 0.
Because the ratios for tangent and secant have "x" as the denominator, they will have asymptotes in the same general area: r/x (sec) and y/x (tan). Likewise for cosecant and cotangent: r/y (cos) and x/y (cot).


BQ #3:
Tangent?: The trig ratio for tangent is sin/cos. When graphed, sin and cosine are both positive in the first quadrant, so when applied to the trig ratio for tangent, sin/cos will actually translate into (+)/(+), which makes tangent in the first quadrant segment of the graph positive. We then use this method for the rest of the quadrants to figure out where tangent is positive or negative. We can find the asymptotes wherever cosine is 0, which is when it intersects with the x-axis.

Cotangent?: Because the trig ratios for cotangent are the opposite of tangent's trig ratio, the concept is basically the same. We use the same method of figuring out whether sine and cosine are positive or negative in a quadrant segment in order to determine whether cotangent is positive or negative as well. Because cotangent and tangent are reciprocals of each other, their trig graphs will be opposites of each other: where tangent goes up, cotangent goes down, and vice versa. The asymptotes for a cotangent graph will occur wherever sin(x)=0.

Secant?: Secant and cosine are reciprocals of each other also, so whenever cosine is positive, secant will also be positive, and whenever cosine is negative, secant will be negative as well. Wherever cosine equals 0, there will be an asymptote. In just one period, there is a half parabola facing upwards where there is a "mountain" and a downwards facing parabola where there is a "valley".

Cosecant?: Cosecant is the reciprocal if sine, so we use similar methods as the previous ones. Because sine is positive in the first two quadrants, cosecant will form a full parabola facing upwards in the first two quadrants. Because sine is negative in the third and fourth quadrants, cosecant will form a full parabola facing downwards in the last two quadrants of the period. Therefore, a whole period will actually include two parabolas: one going up and the other going down.

BQ #4:
Because tangent and cotangent are reciprocals of each other, they will be positive and negative in the same quadrants, yet their asymptotes will be different, allowing for them to be different when graphed. Tangent has asymptotes when x=0 whereas cotangent has asymptotes when y=0. Because tangent and cotangent don't have the same variable as 0, when graphed, they will be different, even though the same segments on the graph will be positive and negative.