# Sine Law

We have been always applying trigonometric ratios to right-angled triangles. Here we shall use the same trigonometric ratios for oblique triangles also.

In this mini-lesson, we shall explore the topic of sine law,* *by finding answers to questions like what is law of sines, what are the applications of sine law, and proof of the sine law.

**Lesson Plan**

**What Is 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.

**Formula**

The formula for sine law is:

\[\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC}\]

- 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.

**Proof of the Sine Law**

To prove the sine law, we consider these two oblique triangles.

In the first triangle, we have:

\[\begin{align}\dfrac{h}{b} &= SinA\\h&=bSinA\end{align}\]

In the second triangle, we have:

\[\begin{align}\dfrac{h}{a} &= SinB\\h&=aSinB\end{align}\]

Also \( Sin(180^\circ - B) = Sin B \).

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

\[\begin{align}aSinB&=bSinA\\\dfrac{a}{SinA}& = \dfrac{b}{SinB}\end{align}\]

Similarly, we can derive a relation for \(SinA\) and \(SinC\).

\[\begin{align}aSinC&=cSinA\\\dfrac{a}{SinA}& = \dfrac{b}{SinB}\end{align}\]

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

\[\begin{align}\dfrac{a}{SinA}& = \dfrac{b}{SinB} = \dfrac{c}{SinC}\end{align}\]

- 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. \[\text{Area of a triangle}=\dfrac{1}{2}abSinC = \dfrac{1}{2}bcSinA = \dfrac{1}{2}caSinB\]
- Also sine law provides a relationship with the radius R of the circumcircle. \[\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC} = 2R \]

**What Are the Applications of Sine Law?**

The sine law can be applied to calculate:

- The length of the side of a triangle
- The unknown angle of a triangle
- The area of the triangle

COSINE LAW: This proves a relationship between the sides and one angle of a triangle. \[ c^2 = a^2 + b^2 - 2ab \cdot CosC\]

TANGENT LAW: This has been derived from the sine law and it gives the relationship between the sides and angles of a triangle. \[ \dfrac{a - b}{a + b} = \dfrac{Tan\frac{(A - B)}{2}}{Tan\frac{(A + B)}{2}}\]

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

**Solved Examples**

Example 1 |

Given a = 20 units c = 25 units and Angle C = \(42 ^\circ \). Find the angle A of the triangle.

**Solution**

For the given data, we can use the following formula of sine law. \[\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC}\]

\[\begin{align}\dfrac{20}{SinA} &= \dfrac{25}{Sin42^\circ} \\\dfrac{SinA}{20} &= \dfrac{Sin42^\circ }{25}\\ SinA&= \dfrac{Sin42^\circ}{25} \times 20\\ SinA&= \dfrac{Sin42^\circ}{25} \times 20\\SinA&= \dfrac{0.6691}{5} \times 4\\SinA &= 0.8363\\ A &= Sin^{-1}0.8363 \\A &=56.7^\circ \end{align}\]

\(\therefore \angle A = 56.7^\circ \) |

Example 2 |

Two angles and an included side is . \( \angle A = 47^\circ \) and \( \angle B = 78^\circ \) and c = 6.3 units. Find the value of a.

**Solution**

Given \( \angle A = 47^\circ \) and \( \angle B = 78^\circ \)

\[\begin{align} \angle A + \angle B + \angle C &= 180^\circ \\47^\circ + 78^\circ + \angle C&= 180^\circ \\ 125^\circ + \angle C &= 180^\circ \\ \angle C &= 180^\circ - 125^\circ \\ \angle C &= 55^\circ\end{align}\]

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

\[\begin{align}\dfrac{a}{SinA} &= \dfrac{c}{SinC} \\\dfrac{a}{Sin47^\circ} &= \dfrac{6.3 }{Sin 55^\circ}\\ a&= \dfrac{6.3}{Sin 55^\circ} \times Sin 47^\circ\\ a &= \dfrac{6.3}{0.8192} \times 0.7314 \\a &= 5.6 \end{align}\]

\(\therefore a = 5.6 units \) |

**Interactive Questions on Law of Sines**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of sine law. The math journey around sine law starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**FAQs on Sine Law**

### 1. What is meant by law of sines?

Law of sines gives a relationship between the sides and angles of a triangle. \[\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC}\]

### 2. When can we use sine law?

We can use the sine law to find:

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

### 3. What is the sine rule?

The sine rule gives the ratio of the sides and angles of a triangle.

\[\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC}\]

Here a, b, c are the length of the sides of the triangle, and A, B, C are the angles of the triangle.

### 4. What are the different ways to represent sine rule formula?

Sine law can be represented in the following three ways.

- \(\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC}\)
- \(\dfrac{SinA}{a} = \dfrac{SinB}{b} = \dfrac{SinC}{c}\)
- \(\dfrac{a}{b} = \dfrac{SinA}{SinB}; \dfrac{a}{c} = \dfrac{SinA}{SinC};\dfrac{b}{c} = \dfrac{SinB}{SinC}\)

### 5. In which cases can we use the sine law?

The sine law can be used for three purposes.

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

### 6. Can sine law be used on a right triangle?

The sine law can also be used for a right triangle.

### 7. What are the possible criteria for law of sines?

The criteria to use the sine law is to have the following.

A pair of lengths of two sides of a triangle and an angle.

OR

A pair of angles of a triangle and the length of one side.