a. Tangent?
Photocredits: desmos.com
In order to fully understand how all trig functions seen on this graph relate, lets go quadrant by quadrant. In the first quadrant, sin, cos and tan are positive. So, therefore, all of the graphs at this point are above the x-axis showing they are positive. In the second quadrant, it's quite different sin is the only trig function that's positive in this quadrant the other two are negative. In quadrant three, tan is the only positive trig and the other two are negative. Lastly, in quadrant four, cos is the only positive and the other two are negative.
So, what's the big deal with these asymptotes and why are they placed in the places they are placed? Tangent has asymptotes because at certain points such as pi/2(90 degrees) and 3pi/2 (270 degrees) tan's ratio (x/y) is undefined. That is why the tangent graph looks split in between sections, the tangent periods don't touch the asymptotes but do get very close.
b. Co-tangent?
Photocredits: desmos.com
Cotangent's graph is very similar to tangent, positive at all the same parts and negative at all the same parts, except that they go in two different directions. Why? Well, because yes they're inverses of each other but they have asymptotes at different points on the unit circle. In the Unit Circle tan's ratio is undefined at 0, pi (180 degrees) and 2pi (360 degrees). Therefore, the co-tangent graph goes downhill.
c. Secant?
Photocredits: desmos.com
Secant, the beautiful secant. So, what's up with this so-called secant? In the first quadrant, all three, sin, cos and secant are positive. In the second quadrant, sin is the only one positive, the other two are negative. Furthermore, the third quadrant, all (sin, cosine and secant) are negative in this quadrant. Lastly, in the fourth quadrant, both cos and secant are positive, and the only one negative is sin.
The asymptotes for secant are on pi/2(90 degrees) and 3pi/2 (270 degrees). Those are the specific asymptotes for secant because that is where secant's ratio is undefined.
What seems so odd is that the asymptote of secant is connected with cos, why is that? That is because in simple words secant is the inverse of cosine!
d. co-secant?
Photocredits: desmos.com
Co-secant is the inverse of sine. In the first quadrant, all three, sin, cosine and co-secant are positive. In the second quadrant, sin and co-secant are positive and cos is negative. Moreover, in the third quadrant, all the three trig functions are negative. And lastly, in the fourth quadrant cos is the only one positive, the other two (sin, co-secant) are negative.
Much like co-tangent, co-secant has certain asymptotes because that is where its ratio is undefined. Co-secant is undefined at at 0, pi (180 degrees) and 2pi (360 degrees).
And just like secant, co-secant is attached to sin because its the inverse of co-secant!
Thank you for reading my blog. Comments, concerns, questions? Feel free to call me *wink*. Just kidding, comment below..
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