Tangents in Geometry – Definition, Derivation, Applications, and FAQs

# Tangents in Geometry – Definition, Derivation, Applications, and FAQs

## Tangent of a Circle Definition

The tangent of a circle is a line that intersects a circle at a single point and is perpendicular to the radius at that point. A tangent of a circle is a straight line that touches the circle at one point and is perpendicular to the radius at that point. The tangent line can be thought of as the limit of a sequence of secant lines, where the distance between each consecutive secant line gets closer and closer to being zero.

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In Geometry, the tangent is defined as a line touching circles or an ellipse at only one point. Suppose a line touches the curve at P, then the point “P” is called the point of tangency. In other words, it is defined as the line which represents the slope of a curve at that point. The tangent equation in differential geometry can be found using the following procedures:

As we know that the gradient of the curve is equal to the gradient of the tangent to the curve at any point given on the curve. We can find the tangent equation of the curve y = f(x) as follows:

• Find the derivative of gradient function using the differentiation rules
• To calculate the gradient of the tangent, substitute the x- coordinate of the given point in the derivative
• In the straight-line equation (in a slope-point formula), substitute the given coordinate point and the gradient of the tangent to find the tangent equation

## Tangent of a Circle

A circle is also a curve and is a closed two dimensional shape. It is to be noted that the radius of the circle or the line joining the centre O to the point of tangency is always vertical or perpendicular to the tangent line AB at P, i.e. OP is perpendicular to AB as shown in the below figure.

Here “AB” represents the tangent, and “P” represents the point of tangency and “O” is the centre of the circle. Also, OP is the radius of the circle.

## Tangent Meaning in Trigonometry

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 other words, it is the ratio of sine and cosine function of an acute angle such that the value of cosine function should not equal to zero. Tangent function is one of the six primary functions in trigonometry.

The Tangent Formula is given as:

Tan A = Opposite Side/Adjacent side

In terms of sine and cosine, tangent may be represented as:

Tan A = Sin A / Cos A

We know that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side whereas the cosine of the angle is the ratio of the length of the adjacent side to the ratio of the hypotenuse side.

That is, Sin A = Opposite Side/ Hypotenuse Side

Cos A = Adjacent Side/ Hypotenuse Side

Therefore, tan A = Opposite Side/ Adjacent Side

In trigonometry, the tangent function is used to find the slope of a line between the origin and a point representing the intersection between the hypotenuse and the altitude of a right triangle

However, in both trigonometry and geometry, tangent represents the slope of some object. Now let us have a look at the most important tangent angle – 30 degrees and its derivation.

### Tangent Sample Problem

Question:

Find the tangent angle of a right triangle whose adjacent side is 5 cm and the opposite side is 7 cm.

Solution:

Given, adjacent side = 5 cm

Opposite side = 7 cm

Formula to find tangent angle is, tan θ=Opposite Side/Adjacent Side

tan θ = 7 cm/5 cm

tan θ = 1.4

Derivation

Length of Hypotenuse = 2×Length of the measure of the opposite side

Length of Adjacent side= √3/2 × Length of Hypotenuse

Length of Adjacent side= √3/2 × (2×Length of Opposite side)

Length of Adjacent side= (√3/2×2) ×Length of Opposite side

Length of Adjacent side=√3 × Length of Opposite side

1√3=Length of opposite side/length of the adjacent side

Since the ratio is tan30⁰,

tan30⁰ = 1/√3

Similarly, we can find the values of other angles like 45, 60 using this property of right-angled triangles

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