Tangent

The word “tangent” is derived from the Latin word “tangere” which means to touch. Generally, we could say that tangent is the line that intersects the circle exactly at one point on its circumference and never enters the interior of the circle. In differentiation, the tangent line is considered to be one of the most important applications. At one point on the curve, the tangent line comes into contact with the curve. To calculate the tangent line equation, we must first determine the curve's equation and the location where the tangent is drawn. The "point of tangency" is the location where the tangent is drawn. In this article, we are going to discuss tangent meaning in geometry, trigonometry, applications in science and technology and also a few of the most frequently asked questions will also be answered.

What is a Tangent Line?

A tangent line to a curve at a point is a straight line that most closely approximates (or "clings to") the curve near that point. As the second point approaches the first, it can be considered the limiting position of straight lines passing through the given point and a nearby point of the curve. If two curves have the same tangent line at a point, they are said to be tangent. The tangent plane to a point on a surface and two surfaces that are tangent at a point are both defined in the same way. In the right triangle trigonometry, the ratio of the side opposite the angle to the side adjacent is known as the tangent of an angle.

The image given below easily demonstrates the difference between a tangent line and a non-tangent line. It is clearly evident that a tangent line never enters the interior of the circle.

Tangent Meaning in Geometry

The tangent line (or simply tangent) to a plane curve at a particular location is a straight line that "just touches" the curve at that point, according to geometry. It was described by Leibniz as the line joining two infinitely close points on a curve. A tangent to the curve y = f(x) at a point x = c is a straight line that passes through the point (c, f(c)) on the curve and has the slope (m) f'(c), where f' is the derivative of f. Space curves and curves in n-dimensional Euclidean space have a similar definition.

The tangent line is "moving in the same direction" as the curve when it passes through the point where the tangent line and the curve meet, called the point of tangency is the best straight-line approximation tangent to the curve at that moment. The graph of the affine function that best approximates the original function at the given point can also be thought of as the tangent line to a point on differentiable curves. Similarly, the tangent plane to a given point on a surface is the plane that "just touches" the surface at that given point. Tangent space is a generalization of the concept of a tangent, which is one of the most fundamental concepts in differential tangent geometry.

Tangent Meaning in Trigonometry

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in trigonometry.

In other words, it is the ratio of an acute angle's sine and cosine functions such that the cosine function's value does not equal zero. In trigonometry, the tangent function is one of the six key functions.

The Tangent Formula in trigonometry is given as:

Tan A = \[\left(\frac{Opposite ~ side}{Adjacent~Side}\right)\]

The sine of every angle is always equal to the length of the opposite side which is divided by the length of the hypotenuse side, whereas the cosine of an angle is the ratio of the adjacent side and the hypotenuse side.

In terms of sine and cosine, the tangent formula may be given as:

Sin A = \[\left(\frac{Opposite ~ side}{Hypotenuse~Side}\right)\]

Cos A = \[\left(\frac{Adjacent ~ side}{Hypotenuse~Side}\right)\]

As we know Tan = $\left(\frac{Sine}{Cosine}\right)$

Hence Tan A = \[\left(\frac{\left(\frac{Opposite~Side}{Hypotenuse~Side}\right)}{\left(\frac{Adjacent~Side}{Hypotenuse~Side}\right)}\right)\]

The tangent function is used in trigonometry to compute the slope of a line between the origin and a point representing the intersection of a right triangle's hypotenuse and altitude. On the other hand, Tangent represents the slope of an object in both trigonometry and geometry.

What is a Vertical Tangent?

A vertical tangent is a tangent line that is vertical in mathematics, particularly calculus. A function whose graph has a vertical tangent is not differentiable at the point of tangency because a vertical line has an infinite slope.

Osculating plane

The plane spanned by the three different points x(t), x(t+h1), and x(t+h2) on a curve as h1,h2 ->0. Let y be a point on the osculating plane, then,

[(y-x), x’,x”]=0

where [P, Q, R] denotes the scalar triple product The osculating plane passes through the tangent. The intersection of the osculating plane with the normal plane is always called the normal vector.

The Slope of a Tangent Line

To understand the slope of the tangent line , let us first consider a curve which is represented by a function f(x). Next , let us also consider a secant line which passses through two points of the curve P(x0, f(x0)) and Q (x0+h,f(x0+h)) . Do take into account that P and Q are at a distance of H units from each other.

Now by using the slope formula we can find the slope of the secant line .

The slope of the secant line = \[\left(\frac{[f(x_{0}+h)-f(x_{0})])}{x_{0}+h-x_{0}}\right)\]

= \[\left(\frac{f(x_{0}+h)-f(x_{0})}{h}\right)\]

The secant line becomes the tangent line at P if Q comes very close to P (by making h→ 0) and merges with P, as shown in the diagram above. h → 0 can be applied to the slope of the secant line to get the slope of the tangent line at P.

Hence now , the slope of the tangent at P = \[\lim_{h \to \ 0 } \left(\frac{f(x_{0}+h)-f(x_{0})}{h}\right) = 0\]

From the limit definition of the derivative or the first principles we know that the above equation is nothing but the derivative of f(x) at x = x0

Now , slope of the tangent at P = f '(x₀)

Therefore from the above explanation it is clearly evident that the slope of the tangent is nothing but the derivative of the function at the point where it has been drawn.

Tangent Line Equation

The point slope form : y - y₀ = m (x - x₀) is used to find the equation of a line with slope ‘m’ which passes through the point (x₀, y₀). Let us now consider the tangent line that is drawn to a curve y = f(x) at a point (x₀, y₀).

We know that the slope of the tangent line , m =(f’(x)) at (x₀, y₀)

Now to get the tangent line equation we just have to substitute m , x₀ and y₀ values in the point slope form , y - y₀ = m (x - x₀).

Hence the tangent line formula is y - y₀ = $f'(x)_{(x_{0},y_{0})}$(x - x₀)

How to Find the Tangent Line Equation

There are a few steps that need to be followed to find the tangent line equation of a curve y = f(x) which is drawn at a point (x₀, y₀)

• If it is said that the tangent is drawn at x = x₀ and if the y coordinate of the point is not given then we need to find the y-coordinate by substituting it in the function y = f(x).

• The next step is to find the derivative of the function y = f(x) and then represent it by f'(x)

• Now to find the slope of the tangent (m ), we need to substitute the point (x₀, y₀) in the derivative f'(x).

• The last step is to find the equation of the tangent using the point-slope form y - y₀ = m (x - x₀).

Exemplar Problem With Step by Step Solutions

Example 1: Point (1,3) lies on a curve given by y = f(x)= x³-x+4 . Find the tangent line's equation.

To write the equation of a tangent line we need the below things:

1. Slope

2. A point on the line

Solution : It is given that the curve contains a point (1, 3)

The slope of the curve is the same as the slope at x = 1 which is equal to the functions derivative at that given point:

f(x) = x3-x+4

f’(x) = 3x2-1

f’(1) = 3(1)-1

f’(1) = 2

Substituting the value of slope (m) in the point-slope form of the line,

y-y0 = m(x-x0)

y-3 = 2(x-1)

Converting the equation into y-intercept form as

y-3 = 2(x-1)

y-3 = 2x-2

y= 2x+1

∴ The equation for the tangent line is y = 2x + 1.

Example 2 : Find the equation of the tangent line of the curve y = 3x2 - 4x at x = -1.

Solution:

The given curve is, f(x) = 3x2 - 4x

Its derivative is f'(x) = 6x - 4.

The slope of the tangent is, m = f'(x) = f'(-1) = 6(-1) - 4 = -10.

The point at which the tangent is drawn is, (x₀, y₀) = (-1, f(-1)) = (-1, 3(-1)2 - 4(-1)) = (-1, 7).

The equation of the tangent line is given by y - y₀ = m (x - x₀)

y - 7 = -10 (x - (-1))

y - 7 = -10 (x + 1)

y - 7 = -10x - 10

y = -10x - 3

Hence the equation of the tangent line of the curve y = 3x2 - 4x at x = -1 is y = -10x - 3

Tangent Applications in Science and Technology

Tangent has a wide range of applications in science and technology because it is a function of both Sine and Cosine functions. Trigonometric functions are widely employed in the fields of engineering and physics. When there is something in a circle shape or something that resembles a circular shape, the sine, cos, and tan functions are expected to appear in the description. The following are some examples of notions that make use of trigonometric functions:

• Artificial Neural Networks

• Empirical Formula and Heuristic functions

• Visualizations (Example: Andrews Plot)

• The behavior of Elementary Particles

• Study of waves like Sound waves, electromagnetic waves

Tangent Properties

• A tangent will touch a curve at only one point.

• A tangent line can never enter the interior of the circle and if it does then it's not a tangent line.

• At a right angle, the tangent touches the radius of the circle at the point of tangency.

Conclusion

A tangent is a line that touches a circle or an ellipse at only one point in geometry. If a line intersects a curve at Q, the point "Q" is referred to as the point of tangency. To put it another way, it's the line that represents the slope of a curve at that point. Throughout the article we have discussed tangent, its definition in geometry, trigonometry, its application in science and technology and also a few problems were solved as well. Students can visit the official website of Vedantu where we have provided links to topics related to tangent.