## Law of Sines

The **Law of Sines** is the relationship between the sides and angles of non-right (oblique) triangles . Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.

To use the Law of Sines you need to know either two angles and one side of the triangle (AAS or ASA) or two sides and an angle opposite one of them (SSA). Notice that for the first two cases we use the same parts that we used to prove congruence of triangles in geometry but in the last case we could not prove congruent triangles given these parts. This is because the remaining pieces could have been different sizes. This is called the ambiguous case and we will discuss it a little later.

## The Ambiguous Case

If two sides and an angle opposite one of them are given, three possibilities can occur.

(1) No such triangle exists.

(2) Two different triangles exist.

(3) Exactly one triangle exists.

There are two angles between 0° and 180° whose sine is approximately 0.5833, are 35.69° and 144.31° .

If we are given two sides and an included angle of a triangle or if we are given 3 sides of a triangle, we cannot use the Law of Sines because we cannot set up any proportions where enough information is known. In these two cases we must use the Law of Cosines .

## Sure … ?

Well, let’s do the calculations for a triangle I prepared earlier:

## How Do We Use It?

Let us see an example:

## Finding an Unknown Angle

In the previous example we found an unknown side …

… but we can also use the Law of Sines to find an **unknown angle**.

In this case it is best to turn the fractions upside down (**sin A/a** instead of **a/sin A**, etc):

## Sometimes There Are Two Answers !

There is one **very** tricky thing we have to look out for:

Two possible answers.

This only happens in the “Two Sides and an Angle **not** between” case, and even then not always, but we have to watch out for it.

Just think “could I swing that side the other way to also make a correct answer?”

But wait! There’s another angle that also has a sine equal to 0.9215…

**The calculator won’t tell you this** but sin(112.9°) is also equal to 0.9215…

So, how do we discover the value 112.9°?

Easy … take 67.1° away from 180°, like this:

180° − 67.1° = 112.9°

So, always check to see whether the alternative answer makes sense.

- … sometimes it will (like above) and there are
**two solutions** - … sometimes it won’t (see below) and there is
**one solution**

## What is Law of Sines?

The law of sines relates the ratios of side lengths of triangles to their respective opposite angles. This ratio remains equal for all three sides and opposite angles. We can therefore apply the sine rule to find the missing angle or side of any triangle using the requisite known data.

### Law of Sines: Definition

The ratio of the side and the corresponding angle of a triangle is equal to the diameter of the circumcircle of the triangle. The sine law is can therefore be given as,

a/sinA = b/sinB = c/sinC = 2R

- Here a, b, c are the lengths of the sides of the triangle.
- A, B, and C are the angle of the triangle.
- R is the radius of the circumcircle of the triangle.

## Law of Sines Formula

The law of sines formula is used for relating the lengths of the sides of a triangle to the sines of consecutive angles. It is the ratio of the length of the side of the triangle to the sine of the angle thus formed between the other two remaining sides. The law of sines formula is used for any triangle apart from SAS triangle and SSS triangle. It says,

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

where,

- a, b, and c are the lengths of the triangle
- A, B, and C are the angles of the triangle.

This formula can be represented in three different forms given as,

- a/sinA = b/sinB = c/sinC
- sinA/a = sinB/b = sinC/c
- a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

**Example: Given a = 20 units c = 25 units and Angle C = 42º. Find the angle A of the triangle.**

**Solution:**

For the given data, we can use the following formula of sine law: a/sinA = b/sinB = c/sinC

⇒ 20/sin A = 25/sin 42º

⇒ sin A/20 = sin 42º/25

⇒ sin A = (sin 42º/25) × 20

⇒ sin A = (sin 42º/25) × 20

⇒ sin A = (0.6691/5) × 4

⇒ sin A = 0.5353

⇒ A = sin^{-1}(0.5363)

⇒ A = 32.36º

**Answer: ∠A = 32.36º**

## Proof of Law of Sines Formula

The law of sines is used to compute the remaining sides of a triangle, given two angles and a side. This technique is known as triangulation. It can also be applied when we are given two sides and one of the non-enclosed angles. But, in some such cases, the triangle cannot be uniquely determined by this given data, called the ambiguous case, and we obtain two possible values for the enclosed angle. To prove the sine law, we consider two oblique triangles as shown below.

In the first triangle, we have:

h/b = sinA

⇒ h = b sinA

In the second triangle, we have:

h/a = sinB

⇒ h = a sinB

Also, sin(180º – B) = sinB

Equalizing the h values from the above expressions, we have:

a sinB = b sinA

⇒ a/sinA = b/sinB

Similarly, we can derive a relation for sin A and sin C.

asinC = csinA

⇒ a/sinA = c/sinC

Combining the above two expressions, we have the following sine law.

a/sinA = b/sinB = c/sinC

**Tips and Tricks on Law of Sines**

- The triangulation technique is used to find the sides of a triangle when two angles and one side of a triangle is known. For this the sine law is helpful.
- This sine law of trigonometry should not be confused with the sine law in physics.
- Further deriving from this sine law we can also find the area of an oblique triangle.

Area of a triangle = (1/2) ab sinC = (1/2) bc sinA = (1/2) ca sinB - Also sine law provides a relationship with the radius R of the circumcircle,a/sinA = b/sinB = c/sinC = 2R
- Cosine law: This proves a relationship between the sides and one angle of a triangle,c
^{2}= a^{2}+ b^{2}– 2ab⋅cos C - Tangent law: This has been derived from the sine law and it gives the relationship between the sides and angles of a triangle.

## Applications of Sine Law

The law of sines finds application in finding the missing side or angle of a triangle, given the other requisite data. The sine law can be applied to calculate:

- The length of the side of a triangle using ASA or AAS criteria.
- The unknown angle of a triangle.
- The area of the triangle.

### Ambiguous Case of Law of Sines

While applying the law of sines to solve a triangle, there might be a case when there are two possible solutions, which occurs when two different triangles could be created using the given information. Let us understand this ambiguous case while solving a triangle using Sine law using the following example.

**Example: If the side lengths of △ABC are a = 18 and b = 20 with ∠A opposite to ‘a’ measuring 26º, calculate the measure of ∠B opposite to ‘b’?**

**Solution:**

Using the sine rule, we have sinA/a = sinB/b = sin26º/18 = sin B/20.

⇒ sin B = (9/10) sin26º or B ≈ 29.149º.

However, note that sin x = sin(180º – x). **∵** A + B < 180º and A + (180º – B) < 180º, another possible measure of B is approximately 180º – 29.149º = 150.851º.

**Think out of the box:**

Find the angles of a triangle if the sides are 12 units, 8.5 units, and 7.2 units respectively.

## Examples Using Law of Sines

**Example 1: Two angles and an included side is∠A = 47º and ∠B = 78º and c = 12.6 units. Find the value of a.**

**Solution:**

Given: ∠A = 47º and ∠B = 78º

∠A + ∠B + ∠C = 180º

⇒ 47º + 78º + ∠C = 180º

⇒ 125º + ∠C = 180º

⇒ ∠C = 180º – 125º

⇒ ∠C = 55º

We shall apply the sine law to find the side of the triangle.

a/sin A = c/sin C

⇒ a/sin 47º = 12.6/sin 55º

⇒ a = 5.62

**Answer: a = 11.24 units **

**Example 2: **It is given ∠A = 47º, ∠B = 78º, and the side c = 6.3. Find the length a.

**Solution:**

To find: Length of a

Given:

∠A = 47º, ∠B = 78º, and c = 6.3.

Since, the sum of all the interior angles of the triangle is 180^{∘, }

Therefore,

∠A + ∠B + ∠C=180º

⇒ 47º + 78º + ∠C = 180º

⇒ ∠C = 55º

Using law of sines formula,

a/sinA = b/sinB = c/sinC

⇒ a/sinA = c/sinC

⇒ a/sin47º = 6.3 / sin55º

⇒ a = 6.3 / sin55º × sin47º

⇒ a = 5.6

**Answer:** a = 5.6

**Example 3: For a triangle, it is given a = 10 units c = 12.5 units and angle C = 42º. Find the angle A of the triangle.**

**Solution:**

To find: Angle A

Given:

a = 10, c = 12.5, and angle C = 42º.

Using law of sines formula,

⇒ a/sinA = b/sinB = c/sinC

⇒ 10/sinA = 12.5/sin 42º

⇒ sin A = 0.5353

⇒ ∠A = 32.36º

**Answer:** **∠A = 32.36º**

## FAQs on Sine Law

### What is Meant by Law of Sines?

The Law of sines gives a relationship between the sides and angles of a triangle. The law of sines in Trigonometry can be given as, a/sinA = b/sinB = c/sinC, where, a, b, c are the lengths of the sides of the triangle and A, B, and C are their respective opposite angles of the triangle.

### When Can We Use Sine Law?

Sine law finds application in solving a triangle, which means to find the missing angle or side of a triangle using the requisite given data. We can use the sine law to find,

- Side of a triangle
- The angle of a triangle
- Area of a triangle

### What is the Sine Rule Formula?

The sine rule formula gives the ratio of the sides and angles of a triangle. The sine rule can be explained using the expression, a/sinA = b/sinB = c/sinC. Here a, b, c are the length of the sides of the triangle, and A, B, C are the angles of the triangle.

### What are the Different Ways to Represent Sine Rule Formula?

Sine law can be represented in the following three ways. These three forms are as given below,

- a/sinA = b/sinB = c/sinC
- sinA/a = sinB/b = sinC/c
- a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

### In Which Cases Can We Use the Sine Law?

The sine law can be used for three purposes as mentioned below,

- To find the length of sides of a triangle
- To find the angles of the triangle
- To find the area of the triangle

### Can Sine Law be Used on a Right Triangle?

The sine law can also be used for a right triangle. sine law can be used in oblique(non-right) as well as in a right triangle to establish a relationship between the ratios of sides and their respective opposite angles.

### What are the Possible Criteria for Law of Sines?

The criteria to use the sine law is to have one of the following sets of data known to us,

- A pair of lengths of two sides of a triangle and an angle.
- A pair of angles of a triangle and the length of one side.

### Does Law of Sines Work With 90 Degrees?

Yes, the law of sines and the law of cosines can be applied to both the right triangle and oblique triangle or scalene triangle to solve the given triangle.

### What is the Law of Sines Ambiguous Case?

The law of sines ambiguous case is the case that occurs when there can be two possible solutions while solving a triangle. Given a general triangle, the following given conditions would need to be fulfilled for the ambiguous case,

- Only information given is the angle A and the sides a and c.
- Angle A is acute, i.e., ∠A < 90°.
- Side a is shorter than side c ,i.e., a < c.
- Side a is longer than the altitude h from angle B, where h = c sin A, .i.e., a > h.

### What are the Applications of the Law of Sines?

The law of sines can be applied to find the missing side and angle of a triangle given the other parameters. To apply the sine rule, we need to know either two angles and one side of the triangle (AAS or ASA) or two sides and an angle opposite one of them (SSA).

## Law of Sines

## Law of Sines Definition

In general, the law of sines is defined as the ratio of side length to the sine of the opposite angle. It holds for all the three sides of a triangle respective of their sides and angles.

So, we use the Sine rule to find unknown lengths or angles of the triangle. It is also called as **Sine Rule**, **Sine Law** or **Sine Formula. **While finding the unknown angle of a triangle, the law of sines formula can be written as follows:

(Sin A/a) = (Sin B/b) = (Sin C/c)

In this case, the fraction is interchanged. It means that Sin A/a, instead of taking a/sin A.

## Law of Sines Proof

We need a right-angled triangle to prove the above as the trigonometric functions, which are mostly defined in terms of this type of triangle only.

**Given:** △ABC, AB = c, BC = a and AC = b.

**Construction:** Draw a perpendicular, CD ⊥ AB. Then CD = h is the height of the triangle. “h” separates the △ ABC in two right-angled triangles, △CDA and △CDB.

**To Show:** a / b = Sin A / Sin B

**Proof:** In the △CDA,

Sin A= h/b

And in △CDB,

Sin B = h/a

Therefore, Sin A / Sin B = (h / b) / (h / a)= a/b

And we proved it.

Similarly, we can prove, Sin B/ Sin C= b / c and so on for any pair of angles and their opposite sides.

## Sine Formula

The formulas used with respect to the law of sine are given below.

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 |

It denotes that if we divide side a by the Sine of ∠A, it is equal to the division of side b by the Sine of∠ B and also equal to the division of side c by Sine of ∠C (Or) The sides of a triangle are to one another in the same ratio as the sines of their opposite angles.

Here, Sin A is a number and a is the length.

## Applications of Sine Law

The trigonometry ratios such as sine, cosine and tangent are primary functions that are used to find the unknown angles or sides of a right triangle. The applications of sine law are given below:

- It can be used to compute the other sides of a triangle when two angles and one side is given.
- When two sides and one non-included angle are given.

The sine law is used to find the unknown angle or unknown side.

As per the law, we know, if a, b and c are the lengths of three sides of a triangle and ∠A, ∠B and ∠C are the angles between the sides, then;

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

Now if suppose, we know the value of one of the sides, and the value of two angles, such as:

a = 7 cm, ∠A = 60°, ∠B = 45°

find b?

By the sine law, we know;

a/sin A = b/sin B

Now putting the values, we get;

7/sin 60° = b/sin 45°

7/(√3/2) = b/(1/√2)

14/√3 = √2 b

b = 14/(√3√2) = 14/√6

## Law of Sines Uses

The law of sines is generally used to find the unknown angle or side of a triangle. This law can be used if certain combinations of measurement of a triangle are given.

- ASA Criteria: Given two angles and included side, to find the unknown side.
- AAS Criteria: Given two angles and a non-included side, to find the unknown side.

These two criteria will provide a unique solution, as AAS and ASA methods are used to prove the congruence of triangles.

**Ambiguous Case:**

In an ambiguous case, if in a triangle two sides and the angle opposite to them are known, then there could be three possibilities:

- That there is no such triangle
- There are two different triangle
- Or there is exactly one such triangle

### Law of sines in Real life

- In real life, the law of sines is used in engineering, to measure the angle of tilt.
- It is used in astronomy, to measure the distance between planets or stars
- Also, the measurement of navigation is possible using the law of sines

Law of sines | Law of cosines |

It is used when we are given with: two angles and one side, or two sides and a non-included angle | It is used when we are given with: three sides, or two sides and the included angle. |

## Solved Example

**Question: Solve △PQR in which ∠ P = 63.5° and ∠Q = 51.2° and r = 6.3 cm.**

**Solution:**

Let’s Calculate the third angle-

∠R = 180° – 63.5°- 51.2 °= 65.3°

Now, Let’s calculate the sides:

6.3/Sin 65.3 = p/Sin 63.5

p = (6.3 × Sin 63.5)/Sin 65.3

p= 6.21 cm approximately.

Similarly, 6.3/Sin 65.3 = q/Sin 51.2

q = (6.3 × Sin 51.2) / Sin 65.3

q= 5.40 cm

∠R= 65.3°

p=6.21 cm

q=5.40 cm

## Frequently Asked Questions on Law of sines

### What is meant by the law of sines?

The law of sine or the sine law states that the ratio of the side length of a triangle to the sine of the opposite angle, which is the same for all three sides. It is also known as the sine rule.

### Write down the sine rule.

If a, b, and c are the sides of a triangle, and A, B, and C are the angles, then the sine rule or the law of sine is given by

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

### Why do we use the law of sines?

The law of sines is used to determine the unknown side of a triangle when two angles and sides are given.

### In which cases, can we use the law of sines?

Generally, the law of sines is used to solve the triangle, when we know two angles and one side or two angles and one included side. It means that the law of sines can be used when we have ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side) criteria.

### Mention the other way to represent the sine rule formula.

The other way to represent the sine law or sine rule is a: b: c = Sin A: Sin B: Sin C.

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