# What is Tangent: Definition and 1000 Discussions

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as the graph of the affine function that best approximates the original function at the given point.Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
The word "tangent" comes from the Latin tangere, "to touch".

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1. ### Find Values of Osculating Circle Tangent to a Parabola

I'm at a loss as to how they got to certain steps in the solutions manual. Here's how far I got with this: Since the circle is tangent to y = x^2 + 1, the slope at (1, 2) is going to be 2, as is the slope of the 2nd derivative of the circle, so then... The derivative of the circle would be...
2. ### Quadratics Having a Common Tangent

I do know that since they have a common tangent line, that means: 2x+a = c-2x Since they both have the point (1,0), then since both equations should equal 0 when x = 1: c(1)-(1)^2 = 0 --> c = 1 So now, I replace c with 1 to solve for a in the two derivatives that are equal (common...
3. ### At what ##x## value is the tangent equally inclined to the given curve?

I had to look this up; will need to read on it. from my research, https://byjus.com/question-answer/the-equation-of-straight-line-equally-inclined-to-the-axes-and-equidistant-from-the-points-1-2-and-3-4-is-ax-by-c-0-where/ ... I have noted that at equally inclined; the slope value is ##1##...
4. ### Separable first order ODE involving tangent

By inspection, we see that ##x=k\pi## is a solution for ##k\in\mathbb Z##. Moreover, the equation implicitly assumes ##x\neq n\pi/2## for odd ##n\in\mathbb Z##, since ##\tan x## isn't defined there. So suppose ##x\neq k\pi##, i.e. ##\tan x\neq 0##, then rearranging and writing ##\tan...
5. ### A About computing the tangent space at 1 of certain lie groups

Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...
6. ### Find the equation of the tangent in the given problem

The textbook solution is indicated below; My approach on this question is as follows; ##\dfrac{dy}{dx} = -\dfrac {8}{x^3}## ##\dfrac{dy}{dx} (x=a) = -\dfrac {8}{a^3}## The tangent equation is given by; ##y= -\dfrac {8}{a^3}x+\dfrac{12+a^2}{a^2}## when ##x=0##, ##y=\dfrac{12+a^2}{a^2}##...
7. ### Find the two points on the curve that share a tangent line

IMPORTANT: NO CALCULATORS I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did f'(a) = f'(b): 1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1 2) 4a^3 - 4a = 4b^3 - 4b 3) 4(a^3 - a) = 4(b^3 - b) 4) a^3 - a = b^3 - b 5) a^3 - b^3 = a - b 6) (a...
8. ### Precise definition of tangent line to a curve

How do we define tangent line to curve accurately ? I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...
9. ### I Carroll GR: Tangent Space & Partial Derivatives

He draws an n-manifold M, a coordinate chart φ : M → Rn, a curve γ : R → M, and a function f : M → R, and wants to specify ##\frac d {d\lambda}## in terms of ##\partial_\mu##. ##\lambda## is the parameter along ##\gamma##, and ##x^\mu## the co-ordinates in ##\text{R}^n##. His first equality is...
10. ### How to find the straight tangent line?

I have solved the gradient: gradf(2,-1)=(4,2) and have the tangent plane: 4x+2y+3=0 Somehow the answer is: 3=2x+y And i really don´t understand why.
11. ### MHB Tikz for f(x)=5x^2-2x and tangent line at 1,3

ok I am actually trying to plot $f(x)=5x^2-2x$ with the tangent line going thru $(1,3)$ which is $8\left(x-1\right)+3$ I thot I could just change this from an example but does seem to like it stack exchange had some samples but they got very complex with other features added anyway mahalo...
12. ### Solve the attached problem that involves circle and tangent

Find Mark scheme here; Find my approach here...more less the same with ms...if other methods are there kindly share... part a (i) My approach is as follows; ##x^2+y^2-10x-14y+64=0 ##can also be expressed as ##(x-5)^2+(y-7)^2=10## The tangent line has the equation, ##y=mx+2## therefore it...
13. ### Find the radius of the smaller circle in the tangent problem

Find the question here and the solution i.e number 10 indicated as ##6-4\sqrt{2}##, I am getting a different solution, my approach is as follows. I made use of pythagoras theorem for the three right angle triangles as follows, Let radius of the smaller circle be equal to ##c## and distance...
14. ### I What is the construction of charts for the tangent bundle on the unit circle?

Hello there! Reading the textbook on differential geometry I didn't get the commentary. In Chapter about vector bundles authors provide the following example Let ##M=S^1## be realized as the unit circle in ##\mathbb{R}^2##. For every ##x\in S^1##, the tangent space ##T_x S^1## can be identified...
15. ### Determine the unit tangent vector

I need a justification that ##|\dfrac {dr}{dt}|##=##\dfrac {ds}{dt}## cheers guys... all the other steps are easy and clear to me...
16. ### I Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question. I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM...
17. ### Multiple angles : Reducing the sum

Problem Statement : Let me copy and paste the problem as it appears in the text : Attempt : I haven't been able to make any significant attempt at solving this problem, am afraid. I tried to reduce all the higher submultiple angles ##2\theta, 4\theta, 8\theta## into ##\theta##, but the...
18. ### Solving for Tangent Lines: Analytical and Graphical Approaches

I did it graphically by using GeoGebra. My question is that what can I do to solve it analytically/algebraically. I used the point-slope formula and obtained $$\frac {y - (a^2-4)} {x - a} = 2a$$, which implies that ##y = (2a)x + (-a^2-4)##. I am not sure how to proceed from here onwards...
19. ### What is a tangent? (multiple meanings)

how many different meanings do you guys know, towards the word "tangent"? in science, I already know 2 meanings: the functions similar to sin, cos, cot, and it is tan; it means two geometry objects are intersected, and they touch each other closely. so I wonder if the different meanings would...
20. ### Finding value of k so two curves are tangent

I tried to equate the derivative of the two equations: $$\cos x=-ke^{-k}$$ Then how to continue? Is this question can be solved? Thanks
21. ### MHB Equation for tangent of the curve

Can anyone help me to find the equation of the tangent to the curve x = 2 cos t, y= 2 sin t where t= pi/3??

27. ### A geometry question in Comsol (draw a semi-circle tangent to a line)

I am new to Comsol. I want to draw my model which is a Tesla valve. The geometry is little complicated and I don't know how to draw a semi-circle tangent to a line. Is it possible? I draw it in Solidworks and imported it into Comsol but it gives error and I think it is better to draw inside...
28. ### Tangent vector fields and covariant derivatives of the 3-sphere

This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated. (a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...
29. ### A multivariate function of Toruses - tangent vectors

Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c. I think this map multiplies tangent vectors by a factor of...
30. E

### B Just a question about the tangent basis

I was reading these notes and then on page 23 I saw something a bit weird. Back in this thread I learned that ##\{ \partial_i \}## form a basis of ##T_p M##, and that a tangent vector can be written ##X = X^i \partial_i##, and it's not too difficult to show that components transform like...
31. ### Calculate the dual basis and tangent basis vectors

a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)? b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...
32. ### I Fixing orientation by fixing a frame in a tangent space

I would like to show that fixing the orientation of k-manifold smooth connected ##S## in ##\mathbb {R} ^ n ## is equivalent to fixing a frame for one of its tangent spaces. What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with...
33. ### MHB Why is the Law of Tangent rarely used or taught in typical trigonometry courses?

The next chapter in my personal studies is Law of Sines and Law of Cosines. I am not there yet. I know there is a Law of Tangent but it is rarely used and never taught in a typical trig course anywhere. Why is the Law of Tangent ignored?
34. ### A Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
35. ### Intersection of a tangent of a hyperbola with asymptotes

Summary:: Question: Show that the segment of a tangent to a hyperbola which lies between the asymptotes is bisected at the point of tangency. From what I understand of the solution, I should be getting two values of x for the intersection that should be equivalent but with different signs...
36. ### Solve this differential equation for the curve & tangent diagram

Here is my attempt at a solution: y = f(x) yp - ym = dy/dx(xp-xm) ym = 0 yp = dy/dx(xp-xm) xm=ypdy/dx + xm xm is midpoint of OT xm = (ypdy/dx + xm) /2 Not sure where to go from there because the solution from the link uses with the midpoint of the points A and B intersecting the x-axis...
37. ### I Product of distances from foci to any tangent of an ellipse

As part of the final stage of a problem, there is some algebraic manipulation to be done (from the solution manual): But I'm getting lost somewhere: Also a bit of general advice needed: This is part of a self-study Calculus course, and I often have difficulty with bigger algebraic...
38. ### B Components of Tangent Space Vector on Parametrized Curve

I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'. To give some background, I'm aware that basis vectors in tangent space are given by...
39. ### I Tangent space basis vectors under a coordinate change

I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: \mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...

44. ### Finding tangent lines for 𝑓(𝑥) = 𝑥^3 − 𝑥 + 6

not quite sure if this is right.. can someone confirm?
45. ### Tangent to a parametrized curve

##x(3)=9-6=3##, ##y(3)=27+9=36##. ##\frac{y'(3)}{x'(3)}=\frac{3\times9+3}{2\times3}=\frac{30}{6}=5##. ##y=5(x-3)+36=5x+21##.
46. ### Deriving the Adjoint / Tangent Linear Model for Nonlinear PDE

I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...
47. ### I Visualizing the cotangent space to a sphere

Is it correct to say that: the cotangent is given by the gradients (*) to all the curves passing through a point and it actually spans the same tangent space to a point of a sphere? If you visualize them as geometric planes (**), the cotangent and the tangent spaces are more than isomorphic...
48. ### MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines

$\tiny{3.2.15}$ Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$. $f(x)=\sqrt{x} \quad [0,4]$ Are the secant line and the tangent line parallel...
49. ### A Question about the derivation of the tangent vector on a manifold

I am trying to understand the following derivation in my lecture notes. Given an n-dimensional manifold ##M## and a parametrized curve ##\gamma : (-\epsilon, \epsilon) \rightarrow M : t \mapsto \gamma(t)##, with ##\gamma(0) = \mathbf{P} \in M##. Also define an arbitrary (dummy) scalar field...
50. ### Propagation of uncertainty with tangent function?

For a lab, I needed to calculate the uncertainty of a refractive index that was found using Snell's law. I found an equation online for propagation of error for any general function, which was I thought that since my equation was I could just get rid of the variable y, and have After...