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.
Tangent and cotangent are opposites of each other. Tangent=sin/cos. Whenever cos=0, tan is undefined because there would have to be asymptotes. The graph would go in a positive direction since they are positive in Quadrant 1 and Quadrant 3. Cotangent= cos/sin. Cotangent is positive in Quadrant 1 and Quadrant 3 as well. The asymptote begins at (0,0) where Quadrant 1 is.

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.

Tangent?

Tangent is equal to sin/cos. It is undefined when cos=0. When sin=0, tangent goes through the x-axis since tan=0.

Cotangent?

Cotangent is equalled to cos/sin. In Quadrant 1, t will be positive since the sin and cos graphs are positive. If the sin=1, cotangent will have asymptotes. In Quadrant 2, cos is negative; sin is positive. In quadrant 3, cos is negative; sin is negative. Their signs cancel out to make a positive cotangent. In Quadrant 4, cos is positive; sin is negative. This will give cotangent a negative answer.

Secant?

The reciprocal of secant is 1/cos. If cos=0, sec is undefined and will not touch. However, if the cos=1, sec amplitudes will touch.

Cosecant?

The reciprocal of cosecant is 1/sin. If sin=0, cosecant is undefined and will not touch. However, if sin=1, csc amplitudes will touch.

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.

The Unit Circle has ratios for trig functions. Sin=y/r, cos=x/r, tan=y/x, csc=r/y, sec=r/x, and cot=x/y. The asymptote is only present when the denominator is equal to 0. However, since r is always equal to 1 in the Unit Circle triangles, this will give us an actual number value for sine and cosine. For us to have an asymptote, we would have to have an undefined answer like in the other four trig graphs give us.

BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the Unit Circle?
ANSWER: Trig graphs relate to the Unit Circle because we just have to imagine the Unit Circle being "unwrapped" into a straight line. This would be like the x-axis of the trig graphs. We have to imagine the Unit Circle still having its coordinates.

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

ANSWER: The period of sine and cosine is 2pi because of their positives and negatives. From quadrant one through four, sine is positive, positive, negative, negative. We imagine sine on the unit circle in portions. At 0* it is 0. At 90* it is 1. At 180* it is 0. At 270* it is -1. At 360* it is 0. These are literally the points on the graph that give sine a mountain-to-valley image. For cosine it is reflected, so we start with a valley then turn into a mountain. For tangent and cotangent, the signs from quadrant one to four are positive, negative, positive, negative. Within this "unwrapped unit circle" we only see two periods. However, this kind of graph only shows us half of the period, hence, it is pi.

• Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig functions don’t have amplitudes) relate to what we know about the Unit Circle?

ANSWER: Sine and cosine have amplitudes of one because the Unit Circle's coordinates are at (0,1), (0,-1), (1,0), and (-1,0). If you look at (0,-1) and (0,1), these points are the graph's highest and lowest points on the graph. The amplitude is not going to pass one because that is where it reaches it's highest point, and it is at its lowest point at -1.