Tangents in Geometry

The tangent of a circle is known as a line touching circles or an ellipse at only one point. Imagine, when a line touches the curve at P, then this point “P” is known as the point of tangency. In differential geometry, the tangent equation can be found using the following methods:

So, we know that finding the gradient of the curve is the gradient of the tangent to the curve at any specified point given on the curve. Hence, the tangent equation of the curve y = f(x) is:

• to find the derivative of gradient function through the rules of differentiation. 

• To find the gradient of the tangent, replace the x- coordinate of the given point in the derivative given. 

• In the straight-line equation’s slope -point formula, replace the gradient of the tangent and given coordinate point to find out the tangent equation. 

Tangent of a Circle Definition

A circle is also known as a curve. It is also a closed two-dimensional shape. It is to be observed that the radius of the circle or the line joining the centre O to the point of tangency or the radius of the circle and tangent line are always perpendicular to each other, i.e. OP is perpendicular to XY as shown in the below figure.

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Here “XY” is the tangent of a circle given, and “OP” is the point of tangency and the tangent radius and the point “O” represents the centre of the circle. 

Thus, the radius and the tangent to a circle are related to each other, tangent to a circle formula that can be well explained using the tangent theorem.

Tangent Meaning in Trigonometry

The tangent of an angle is called the ratio of the length of measure of the opposite side to the length of the adjacent side's measure. Hence, it is regarded as the ratio of sine and cosine function of an acute angle; however, the value of cosine function should not equal to zero. It is regarded as one of the six primary functions in trigonometry.

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The direct common tangent formula is:

Tan P = Opposite Side/Adjacent side

Tangent may be given by using Sine and Cosine as:

Tan P = Sin P / Cos P 

The sine of an angle is the length of the measure of the opposite side divided by the length of the hypotenuse side's measure. The cosine of the angle is given by the ratio of the length of the measure of the adjacent side to the ratio of the length of the measure of the hypotenuse side.

So, That is, Sin P = Opposite Side/ Hypotenuse Side

Cos P = Adjacent Side/ Hypotenuse Side

 tan P = Opposite Side/ Adjacent Side

In trigonometry, the tangent function will help find the slope of a line between the point representing the intersection, the hypotenuse and the altitude of a right triangle and the origin. 

Hence, tangent signifies the slope of some object in Trigonometry and tangent geometry. Now let us see at the most important tangent angle – 30 degrees and its derivation.

Derivation of Value of Tangent 30 Degrees

As per the properties of a right-angle triangle when its acute angle equals 30⁰, then the length of the hypotenuse is double the length of the opposite side. The length of the adjacent side is 3√2 times to the length of the measure of the hypotenuse side. 

Hence, 

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.

Applications of Tangents in Science and Technology

Tangent has a wide range of use in science and technology as it is the function of sine and cosine. Some of the areas that use trigonometric functions are the Artificial Neural Networks, visualisations, behaviour of elementary particles, and waves like sound waves, electromagnetic waves.