Law of Tangents formula proof and solved examples
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 in Trigonometry Explained
1. What is the Law of Tangents in trigonometry?
The Law of Tangents states that in any triangle, the ratio of the difference and sum of two sides equals the ratio of the tangents of half the difference and half the sum of their opposite angles. It is written as:
(a − b)/(a + b) = tan[(A − B)/2] / tan[(A + B)/2]
Where:
- a and b are two sides of a triangle
- A and B are the angles opposite those sides
2. What is the formula for the Law of Tangents?
The formula for the Law of Tangents is:
(a − b)/(a + b) = tan[(A − B)/2] / tan[(A + B)/2]
It can also be written as:
- (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. How do you use the Law of Tangents to solve a triangle?
You use the Law of Tangents by substituting known sides and angles into the formula to find an unknown side or angle. Follow these steps:
- Identify two known sides and their opposite angles.
- Substitute into (a − b)/(a + b) = tan[(A − B)/2] / tan[(A + B)/2].
- Solve for the unknown angle or side.
- Use the angle sum property (A + B + C = 180°) if needed.
4. When should you use the Law of Tangents?
The Law of Tangents is used when you know two sides and their opposite angles or when comparing two sides and two angles in an oblique triangle. It is particularly useful:
- In ASA or AAS cases
- When checking solutions from the Law of Sines
- When working with half-angle identities
5. What is an example of the Law of Tangents?
An example of the Law of Tangents is solving for an angle when two sides are known. Suppose:
- a = 7, b = 5
- A + B = 120°
(7 − 5)/(7 + 5) = tan[(A − B)/2] / tan(60°)
This simplifies to:
- 2/12 = tan[(A − B)/2] / √3
- tan[(A − B)/2] = √3/6
6. How is the Law of Tangents different from the Law of Sines?
The Law of Tangents relates the difference and sum of two sides to the tangents of half-angle differences, while the Law of Sines relates each side directly to the sine of its opposite angle. The formulas are:
- Law of Tangents: (a − b)/(a + b) = tan[(A − B)/2] / tan[(A + B)/2]
- Law of Sines: a/sinA = b/sinB = c/sinC
7. Can the Law of Tangents be used for right triangles?
Yes, the Law of Tangents can technically be used for right triangles, but it is rarely necessary. In right triangles, simpler trigonometric ratios like:
- sin θ = opposite/hypotenuse
- cos θ = adjacent/hypotenuse
- tan θ = opposite/adjacent
8. What are the conditions required to apply the Law of Tangents?
To apply the Law of Tangents, you must have a non-degenerate triangle with known side–angle relationships. Specifically:
- The triangle must satisfy A + B + C = 180°
- Two sides and their opposite angles should be known or related
- The denominator (a + b) must not be zero
9. Why is the Law of Tangents not commonly used?
The Law of Tangents is not commonly used because the Law of Sines and Law of Cosines are simpler and more direct. The tangent formula involves:
- Half-angle calculations
- More algebraic manipulation
- Longer computational steps
10. How is the Law of Tangents derived?
The Law of Tangents is derived from the Law of Sines combined with trigonometric identities for sine sums and differences. Starting with:
a/sinA = b/sinB
Rewriting gives:
- a/b = sinA/sinB
- (a − b)/(a + b) = tan[(A − B)/2] / tan[(A + B)/2]
