Law of Sines Formula

Law of Sine Formula is a basic law of trigonometry that defines the relation between the sides and the angles of the triangle. It is used to express the relation between the sides and the angles of the triangle. It is also known as Sine Law, Sine Rule, and Sine Formula.

Let’s learn about the Law of Sine in detail, including its formula, derivation, and examples.

Law of Sine Definition

Law of sine is the ratio of the side length to the sine of the opposite angles. For a triangle with sides, a, b, and c with respective angles, ∠A, ∠B, and ∠C the sine law states that,

a/sin A = b/sin B = c/sin C

Law of Sine Formula

The law of sine formula is used to find the relation between the lengths of a triangle’s sides to the sines of consecutive angles. It is the ratio of the length of the triangle’s side to the sine of the angle generated by the other two remaining sides.

It is given by :

a/sin A = b/sin B = c/sin C

where,

• a, b, and c are Lengths of Triangle
• A, B, and C are Angles of Triangle

Law of Sine Proof

Law of sine is proved using the triangles added below,

To derive Sine Law, consider two oblique triangles illustrated above,

sin A = h/b

=> h = b sin A…(1)

In Right Triangle,

sin B = h/a

=> h = a sin B…(2)

From (1) and (2), we get

a sin B = b sin A

=> a/sinA = b/sinB

Similarly,

a/sinA = c/sinC

By combining the two formulas above,

a/sin A = b/sin B = c/sin C

The is Law of Sine formula.

Sine Formula

Sine Formula is the other name of the Law of Sine, or Sine Rule. The sine formulas are various formulas that are related to the side and angle of the triangle. The formulas listed below are used to solve various trigonometric problems :

a / sin A = b / sin B = c / sin C
a:b:c = sin A: sin B: sin C
a/b = sin A/sin B
b/c = sin B / sin C

Applications of Sine Law

Some of the important applications of sine law are :

• It is used to find the length and angle of a triangle when sides or angles are given.
• It is used to solve various trigonometric problems.
• It is used in finding the height and distances of various real-life cases, etc.

Law of Sines and Cosines

Law of Sine and Cosine relate the sine and cosine of an angle of the triangle with the other angles or sides of the triangle. Let us suppose we are given a triangle ABC with sides a, b, and c and angles A, B, and C respectively then,

Law of Sine is given by :

a/SinA = b/SinB = c/SinC

Law of Cosine is given by,

a² = b² + c² − 2bc cos A

or, b² = a² + c² − 2ac cos B

or, c² = a² + b² − 2ab cos C

Related :

• Triangle
• Sine Rule
• Law of Sine and Cosine
• What is Law of cosine
• Formula for finding Angles

Examples on Sine Law

Example 1: It is given for a triangle ABC, a = 20 units, c = 25 units, and ∠C = 30°. Find ∠A of the triangle.

Solution:

Given,

• a = 20 units
• c = 25 units
• ∠C = 30°

Using Sine Formula

a/sin A = c/sin C

20/sin A = 25/sin 30

sin A = 0.40

A = 23.5°

Example 2: It is given for a triangle ABC, b = 15 units, c = 20 units, and ∠C = 60°. Find ∠B of the triangle.

Solution:

Given,

• b = 15 units
• c = 20 units
• ∠C = 60°

Using Sine Formula

b/sin B = c/sin C

15/sin B = 20/sin 60

sin B = 0.649448

B = 40.5°

Example 3: It is given for a triangle ABC, b = 30 units, c = 40 units, and ∠C = 30º. Find ∠B of a triangle.

Solution:

Given,

• b = 30 units
• c = 10 units
• ∠C = 30°

Using Sine Formula

b/sin B = c/sin C

30/sin B = 40/sin 30

sin B = 0.374607

B = 22°

Law of Sine Practise Problems

Q1. If in a triangle with sides, a = 8, b = 7, and angle ∠A = 120° are given. Find the corresponding value of ∠B.

Q2. In a triangle with sides, a = 12, b = 9, and angle ∠A = 90° are given. Find the corresponding value of ∠B.

Q3. For a triangle of sides a = 6, b = 4, and angle ∠A = 60° are given. Find the corresponding value of ∠B.

Q4. In a triangle of sides, a = 18, b = 12, and angle ∠A = 30° are given. Find the corresponding value of ∠B.

FAQs on Law of Sine

What is Law of Sine in Trigonometry?

Law of Sine in Trigonometry is one of the fundamental law that gives the relation between sides and the angles of a triangle. It is used for finding the unkown angle or side of triangle if other sides of a triangle are given.

What is Law of Sine Formula?

Law of Sine Formula is a Formula that gives the relation between sides and angles of a triangle. For any triangle ABC if the sides of triangles are a, b, and c and the angles of triangles are A, B, and C respectively then law of sine formula is,

a/sinA = b/sinB = c/sinC

What are the Uses of Law of Sine?

Law of Sine is used to find the unknown or missing angle of the triangle if other parameter of the triangle are given. Sine rule is applied when two angles and one side (AAS or ASA) or two sides and one of the opposite angles (SSA) are given.

When can you use the Law of Sine?

Law of Sin is useful for finding a missing angle when an angle and two sides of a triangle are given, or it is used for finding a missing side when given two angles and one side of a triangle are given.

What is sin θ?

Sin θ is one of the six trigonometric ratio that is used to solve various mathematical and physical problems. For a right angle triangle, sin θ is the ratio of its perpendicular and hypotenuse.

Sin θ = Perpendicular(P)/Hypotenuse(H)

What is Law of Sines Ambiguous Case?

Law of sines ambiguous case is the case when the law of sine is not applicable, it occurs when we have two possible solutions while solving a triangle. The condition for the ambiguous case are,

• If we are given a triangle in which angle A and sides a and c
• Where angle A is acute
• Where side a is shorter than side c
• Where side a is longer than the altitude h from angle B, where h = c sin A

Is Sine Law Used on Right Triangles?

Sine Law is easily used in Right Triangles.