Tangent

The word tangent means “To touch”. “Tangere” is the Latin term for the same thing. In general, a tangent is a line that meets the circle precisely at one point on its circumference but never enters the circle’s interior. Many tangents can exist in a circle. They run parallel to the radius. In mathematics, derivatives are frequently used to determine how the value of one variable varies concerning another. Derivates are also observed and discussed in real life. The word speed, for example, is commonly used, but few people realize that it is a derivative; speed is the change in distance vs. time. 

• Determine if a function is rising or decreasing, as well as the period over which it is increasing or decreasing.
• To figure out what several variables’ approximate values are 
• Tangent and normal curves are discovered.
• To find the maximum and lowest values of a function.
• To determine if a function is concave up or down by determining its concavity.
• Use the calculations below to calculate velocity and acceleration from velocity 

Tangent

In geometry, a tangent is a line that is generated from an exterior point and passes through a point on a curve.

Tangent line to the circle:

A circle C is intersected by a tangent line t at a single point T. Secant line, on the other hand, crosses a circle twice, whereas another line may not intersect a circle at all. Many geometrical transformations, such as scaling, rotations, translations, inversions, and map projections, retain this feature of tangent lines. Even if the tangent line and circle may be twisted, these alterations do not change the incidence structure of the line and circle. A tangent to the circle is the line that crosses or intersects the circle at a single point. The point of tangency can be defined as the intersection of a tangent and a circle. The tangent is parallel to the circle’s radius, with which it intersects. Any curved form can be termed a tangent. Because tangent is a line, it has its own equation.

A circle’s radius is perpendicular to the tangent line passing through its terminus on the circumference. A tangent line, on the other hand, is perpendicular to a radius passing through the same endpoint.

Tangent to the curve’s properties:

• The Tangent has the following features:
• A tangent only touches one point on a curve.
• A tangent is a line that does not pass through the circle’s center.
• The tangent forms a right-angle contact with the radius of the circle at the point of tangency.

Half Angle Formula of Tan 

Tangent formulae of half-angle in trigonometry connect the tangent of half an angle to trigonometric functions of the complete angle. The stereographic projection of the circle onto a line is the tangent of half an angle. The following formulae are among them:

the half-angle formula of tan is, 

tan ( θ /2) = [sin ( θ /2)] / [cos ( θ /2)]

From the half-angle formulas of sin and cos,

tan ( ( θ /2plane) = [±√(1 – cos ( θ )/2] / [±√(1 + cos ( θ )/2]

= ±√ [(1 – cos ( θ ) / (1 + cos ( θ )]

This is the formula of tan ( θ /2).

 Let us evaluate the other two formulas by firstly, rationalizing the denominator. We get,

tan ( θ /2) = ±√[(1 – cos ( θ ) / (1 + cos ( θ )] × √[(1 – cos ( θ ) / (1 – cos ( θ )]

= √[(1 – cos ( 2θ ) / (1 – cos2( θ )]

= √[(1 – cos  2θ / sin2 θ]

= (1 – cos θ) / sin θ

This is the 2nd formula of tan (θ /2). 

To evaluate another formula, let us first multiply and divide the above formula by (1 + cos θ). Then we get

tan (θ /2) = [(1 – cos θ) / sin θ] × [(1 + cos θ) / (1 + cos θ)]

tan (θ /2) = (1 – cos2 θ) / [sin θ (1 + cos θ)]

tan (θ /2) = sin2 θ / [sin θ (1 + cos θ)]

tan (θ /2) = sin θ / (1 + cos θ)

Thus, tan (θ /2) = ±√[(1 – cos θ) / (1 + cos θ)] = (1 – cos θ) / sin θ = sin θ / (1 + cos θ).