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.
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