Table of Contents
Introduction to Law of Tangents
A tangent is a straight line that touches a curve at a single point, called the point of tangency. The tangent line is perpendicular to the radius at the point of tangency.
The law of tangents states that the tangent line to a curve at a point is the line that is perpendicular to the radius at the point of tangency.
Tangent Rule Explanation
The tangent rule is a mathematical rule that states that the derivative of the tangent to a curve at a point is the slope of the curve at that point. The tangent rule is important in calculus because it allows for the determination of the slope of a curve at a specific point.
Law of Tangents Proof
The law of tangents states that the tangent to a curve at a given point is the line that is perpendicular to the line joining the point to the curve’s nearest point.
Proof:
Given: A curve and a point on the curve.
To Prove: The tangent to the curve at the given point is the line that is perpendicular to the line joining the point to the curve’s nearest point.
Proof:
Let P be the given point on the curve and let Q be the point on the curve that is closest to P.
Construct a line through P and Q that is perpendicular to the line joining P and Q. This line is the tangent to the curve at P.
Statement of Rule of Tangents
The Rule of Tangents states that the tangent to a curve at a point is the line that is perpendicular to the line joining the point to the curve’s curve.
Law of Tangents Proof
Given: A line and a point not on the line
To prove: The line segment connecting the point to the line is a tangent
Proof:
Let be the line and be the point not on the line.
We will prove that the line segment connecting and is a tangent.
First, we need to show that is perpendicular to .
We can use the formula for the slope of a line to show this:
Since is perpendicular to , we can conclude that the line segment connecting and is a tangent.
Law of Tangent Formula
The law of tangent states that the tangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side.