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
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 the Law of Tangents in trigonometry?
The Law of Tangents, also known as the tangent rule, states the relationship between the sum and difference of two sides of a triangle and the tangents of half the sum and difference of their opposite angles. Specifically, it asserts that the ratio of the difference between any two sides to their sum is equal to the ratio of the tangent of half the difference of the opposite angles to the tangent of half their sum. It is used for solving oblique triangles (non-right-angled triangles).
2. What is the formula for the Law of Tangents?
For any triangle with angles A, B, and C and opposite sides with lengths a, b, and c respectively, the Law of Tangents can be expressed in the following three forms:
- (a - b) / (a + b) = tan[(A - B)/2] / tan[(A + B)/2]
- (b - c) / (b + c) = tan[(B - C)/2] / tan[(B + C)/2]
- (c - a) / (c + a) = tan[(C - A)/2] / tan[(C + A)/2]
3. In which cases can the Law of Tangents be applied to solve a triangle?
The Law of Tangents is primarily used to find the remaining parts of a triangle in the following two cases, as per the CBSE syllabus for 2025-26:
- Side-Angle-Side (SAS): When the lengths of two sides and the measure of the included angle are known. The law helps find the other two angles.
- Angle-Side-Angle (ASA): When two angles and the included side are known. Although the Law of Sines is more direct here, the Law of Tangents can also be used after finding the third angle.
4. How is the Law of Tangents derived using the Law of Sines?
The Law of Tangents can be proven starting from the Law of Sines. Here is the conceptual approach:
- Start with the Law of Sines: a/sin(A) = b/sin(B) = k, where k is a constant.
- Express sides a and b as a = k sin(A) and b = k sin(B).
- Form the ratio (a - b) / (a + b) and substitute the expressions for a and b: [k sin(A) - k sin(B)] / [k sin(A) + k sin(B)].
- Cancel 'k' and apply the sum-to-product trigonometric identities for sin(A) - sin(B) and sin(A) + sin(B).
- After simplification, the expression becomes [2 cos((A+B)/2) sin((A-B)/2)] / [2 sin((A+B)/2) cos((A-B)/2)], which simplifies to tan[(A-B)/2] / tan[(A+B)/2], thereby proving the law.
5. Why is the Law of Tangents used less frequently in exams compared to the Law of Sines and Cosines?
The Law of Tangents is used less often for a few key reasons. The Law of Sines and the Law of Cosines are more direct and versatile, collectively covering all possible cases of solving a triangle (SSS, SAS, ASA, AAS). While the Law of Tangents was historically valuable for calculations involving logarithms before the advent of calculators, modern calculators can handle the computations required by the Law of Cosines just as easily. Therefore, students and mathematicians often prefer the more straightforward application of the Sine and Cosine rules.
6. When is the Law of Tangents a more efficient choice than the Law of Cosines?
The Law of Tangents can be more computationally efficient in the Side-Angle-Side (SAS) case if you need to find the remaining angles directly without first calculating the third side. For instance, if you are given sides 'a', 'b', and the included angle 'C':
- Using the Law of Cosines, you would first find side 'c', and then use the Law of Sines to find another angle.
- Using the Law of Tangents, you can directly find the value of (A - B) since you know (A + B) = 180° - C. This gives you a system of two simple equations to solve for angles A and B simultaneously, which can sometimes involve fewer steps.
7. How does the Law of Tangents for plane triangles relate to Napier's Analogies?
Napier's Analogies, also known as the Rules of Tangents for spherical triangles, are the equivalent of the Law of Tangents but applied to spherical geometry (triangles drawn on the surface of a sphere). While the Law of Tangents deals with plane triangles in Euclidean space, Napier's Analogies adapt this relationship for the curved sides (arcs of great circles) and different angle properties found in spherical triangles. Both sets of laws serve a similar purpose: to solve triangles when sides and angles are known, but they are formulated for different geometric contexts.
