Hi,
When we talk about a line that is tangent to a point on some curve I can't understand the relationship between this tangent and the tangent notion which is in trigonometry (that is sine/cosine or opp/adj). I wonder if there is any relationship between these or not and if there is, how are they related to each other.
I will appreciate it if anyone helps me.
Hello mohammadrezaaalaei!
According to my understanding and what i have leant so far:
A tangent line is a straight line that just barely touches a curve at one point. It is that the tangent line and the curve are both going in the same direction at their point of contact. like for example, having a wavy curve thingy, the tangent line and the curve don't really seem to have much in common because the tangent line is absolutly straight, which they have more in similar.
And in maths explanation is a slope and its point of contact with a curve. A line's slope is its steepness, or rate of change both horizontally and vertically as it travels away from the origin position.
To find the slope of a tangent line, first of all equation's secant line, or a line that connects two points on a curve. To find the equation of a line, we need the slope of that line. With a tangent line, but with a secant line, because we have two points, anyways
The slope of secant line formula is:
slope of the secant line = [f(x + h) - f(x)] /h
and the tangent linr formula is:
y-y1 = m(x-x1),where (x1,y1)is known as the point of the line,as m is the slope of the tangent line. Just try mastering these formula into your work, and you will understand! :)
Hope you understand! feel free to ask questions, and I will be glad to help!
Best Regards!
~Jessica.
Thank you Jessica. I appreciate your explanation and the time you've spent on this comment. your explanation was so clear and understandable. but you know, we have some notions in trigonometry such as tangent (sin(x) / cousin(x)) or secant ( 1 / cos(x)) and we use the same names for these lines touching curve at one point ( tangent line) or crossing curve at two-point ( secant line ) and I cant see the connection between them. But maybe there isn't any relationship between them and they are just names.
I understand your point, but remember about the co-functions, the secant, the co-secant, and the cotangent as you listed?
Remember the co-secant is the reciprocal / reverse of the sine, so wherever the sine is zero, the co-secant will be undefined, so there will be a vertical asymptote, and wherever the sine reaches its maximum value of 1, the co-secant will reach its minimum value of 1 and wherever the sine reaches its minimum value of –1, the co-secant will reach its maximum value of –1. Wherever the sine is positive but less than 1, the co-secant will be positive but greater than 1; wherever the sine is negative but greater than –1, the co-secant will be negative but less than –1. By using the same reasoning with the cosine wave, you can create the secant graph trough the co-secant.
So The Secant Graph:
Just as I explained the co-secant, the secant and co-secant have periods of length 2pi, and we don't consider amplitude for these curves. The cotangent is the reciprocal of the tangent. Wherever the tangent is zero, the cotangent will have a vertical asymptote; wherever the tangent has a vertical asymptote, the cotangent will have a zero. And the signs on each interval will be the same.
And The Cotangent Graph:
The cotangent has a period of "pi", and you don't bother with the amplitude, when you need to do the graphs, you may be tempted to try to compute a lot of plot points, but all you really need to know is where the graph is zero, where it's equal to 1, or where it has a vertical asymptote, If you know the behaviour of the function at zero, pi/2, pi, 3pi/2, and 2pi, then you can fill in the rest, that is all you need to know!
Hopefully you understand!
-Best Regards,
~Jessica.
Thanks for the amazing knowledge .
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