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Law of Tangents

Maths

Law of Tangents

Reviewed by:

Rama Sharma

Introduction to Law of Tangents

Law of tangents is a law in trigonometry which relates the sides and angles of a right triangle. Tangent rule gives the relationship between the sum and differences of the sides and angles of a triangle. The tangent rule can be used to find the remaining parts of any triangle for which two sides and one angle or one side and two angles are given. Law of tangents finds extensive applications in various mathematical computations just like sine and cosine laws. The law of tangents for a triangle with angles A, B and C opposite to the sides a, b and c respectively is given as:

\[\frac{a-b}{a+b}\] = \[\frac{tan(\frac{A-B}{2})}{tan(\frac{A+B}{2})}\]

Tangent Rule Explanation

The rule of tangent establishes a relationship between the sum and differences of any two sides of a triangle and their corresponding angles. The tangent rule states that the ratio of difference and sum of any two sides of a triangle is equal to the ratio of the tangent of half the difference and tangent of sum of the angles opposite to these sides. Rule of tangents can be used to find the unknown parts of a triangle when two sides and an angle or two angles and a side are given.

Law of Tangents Proof

The rule of tangents can be proved using the sine rule. Sine rule states that the ratio of any side of a triangle and the sine of the angle opposite to it is a constant. This basic rule is the foundation for proving the rule of tangents.

Statement of Rule of Tangents

The ratio of sum and difference of any two sides of a triangle is equal to the ratio of tangent of half the sum and tangent of half the difference of the angles opposite to the corresponding sides.

Data

In the triangle ABC, ∠A,∠B and ∠C are the angles opposite to the sides ‘a’, ‘b’ and ‘c’ respectively.

To Prove: What is tangent law?

i.e. \[\frac{a-b}{a+b}\] = \[\frac{tan(\frac{A-B}{2})}{tan(\frac{A+B}{2})}\]

Law of Tangents Proof

Statement	Reason
\[\frac{a}{SinA}\] = \[\frac{b}{SinB}\] = \[\frac{c}{SinC}\]	Applying sine rule to the triangle ABC
\[\frac{a}{SinA}\] = \[\frac{b}{SinB}\] = d	Equating the ratio to a constant
\[\frac{a}{SinA}\] = d and \[\frac{b}{SinB}\] = d	Equating each ratio to the constant ‘k’
a = d Sin A and b = d Sin B	Cross multiplication
a + b = d Sin A + d Sin B = d (Sin A + Sin B) → (1)	Sum of ‘a’ and ‘b’
a - b = d Sin A - d Sin B = d (Sin A - Sin B) → (2)	Difference between ‘a’ and ‘b’
Sin M + Sin N = 2 Sin\[(\frac{M+N}{2})\] Cos\[(\frac{M-N}{2})\] Sin M - Sin N = 2 Cos \[(\frac{M+N}{2})\] Sin \[(\frac{M-N}{2})\]	Trigonometric identities
\[\frac{a-b}{a+b}\] = \[\frac{d(SinA-SinB)}{d(SinA+SinB)}\] = \[\frac{SinA-SinB}{SinA+SinB}\]	Dividing (1) and (2)
\[\frac{2Cos(\frac{A+B}{2})Sin(\frac{A-B}{2})}{2Sin(\frac{A+B}{2})Cos(\frac{A-B}{2})}\]	Substituting the trigonometric identities in the above equation
\[\frac{a-b}{a+b}\] = \[\frac{tan(\frac{A-B}{2})}{tan(\frac{A+B}{2})}\]	By definition, Tan R = Sin R / Cos R

The final equation gives the law of tangent formula.

Law of Tangent Formula

Consider a triangle with sides ‘f’, ‘g’ and ‘h’ opposite to the vertices F, G and H. The sum of two sides is (f + g) or (g + h) or (h + f). Similarly the difference between two sides is given as (f - g) or (g - h) or (h - f).

The law of tangent formula for the ratio of difference and sum of two sides of a triangle is given as:

\[\frac{F-g}{f+g}\] = \[\frac{tan(\frac{F-G}{2})}{tan(\frac{F+G}{2})}\] → (1)

\[\frac{g-h}{g+h}\] = \[\frac{tan(\frac{G-H}{2})}{tan(\frac{G+H}{2})}\] → (2)

\[\frac{h-f}{h+f}\] = \[\frac{tan(\frac{H-F}{2})}{tan(\frac{H+F}{2})}\] → (3)

Fun Facts

Law of tangents for triangles was given by a Persian Mathematician Nasir al-Din al-Tusi in the 13th century. He explained, the law of tangents for spherical triangles.
The spherical law of tangents states that the ratio of tangent of the difference between two sides and the tangent of its sum is equal to the ratio of tangent of the half of the difference between their opposite angles and the tangent of half of their sum. For a triangle with angles P, Q and R opposite to the sides ‘p’, ‘q’ and ‘r’ respectively, the spherical law of tangents is given as:

\[\frac{tan(\frac{a-b}{2})}{tan(\frac{a+b}{2})}\]= \[\frac{tan(\frac{A-B}{2})}{tan(\frac{A+B}{2})}\]

Conclusion

Law of tangents is related to the relationship between the sum and differences of the sides and angles of a triangle. The application of this rule is in finding the remaining parts of a triangle for which two sides and one angle or one side and two angles are given.

FAQs on Law of Tangents

1. What is a tangent and its Applications?

In geometry, tangent is defined as a line touching circles or an ellipse at only one point. The point where it touches the circle is called the point of tangency. The tangent equation in differential geometry can be found using the following procedures:

In trigonometry, the tangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side. It is the ratio of sine and cosine function of an acute angle where the cosine function’s value is not equal to zero.

Tangent of an Angle = \[\frac{Length\:of the leg opposite to the angle}{Length of the leg adjacent to the angle}\]

In trigonometry, the tangent function is one of its six primary functions. Since tangent is the function of both sine and cosine functions, it has a wide range of applications in science and technology. Some of the uses of these trigonometric functions are found in the following areas:

Empirical Formula
Heuristic functions
Study of waves (sound waves, electromagnetic waves)
Visualizations
Artificial Neural Networks
Elementary Particles be

2. What is the law of cotangents?

The law of cotangents is a relationship between the lengths of the sides of a triangle and the cotangents of the halves of the three angles. This is known as the Cot Theorem. Just like the equality of three quantities expressed by the law of sines are equal to the diameter of the circumscribed circle of the triangle, the law of cotangents expresses the relation between three factors: radius of the inscribed circle of a triangle (the inradius), its sides and angles.

3. What is the tangent to a Circle?

Tangent to a circle is the line that touches the circle at only one point. There can be only one tangent at one point to a circle. The point at which the tangent meets the circle is called the point of tangency. If it touches a curve at more than one point then it is considered to be a simple line. Also, the tangent is perpendicular to the radius of the circle. Based on where the point of tangency lies with respect to the circle, there are three conditions:

When the point lies inside the circle
When the point lies on the circle
When the point lies outside the circle

Here are Some Equations of the Tangent to the Circle:

The tangent to a circle equation x²+ y²=a² at (x₁, y₁) is xx₁+yy₁= a²
The tangent to a circle equation x²+ y²+2gx+2fy+c =0 at (x₁, y₁) is xx₁+yy₁+g(x+x₁)+f(y +y₁)+c =0
The tangent to a circle equation x² + y²=a² at (a cos θ, a sin θ ) is x cos θ+y sin θ= a
The tangent to a circle equation x²+ y²=a² for a line y = mx +c is y = mx ± a √[1+ m²]

4. What are the Applications of the law of tangents?

The law of tangent states the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
The law of tangents is also applied to a non-right triangle.
It can be used to find the remaining parts of a triangle if two sides and one angle or two angles and one side are known. These are referred to as side-angle-side (SAS) and angle-side-angle (ASA).

The four cases involved are:

One side and two angles
Two sides and one opposite angle
Two sides and the angle between them
Three Sides