# Law of Sines Calculator

## Solve triangles using the law of sines

The calculator will solve the given triangle using the law of sines (wherever possible), with steps shown.

Related calculator: Law of Cosines Calculator

The Sine Rule, also known as the Law of Sines, describes the relationship between the sides of a triangle and its angles. Our Law of Sines Calculator implements this rule to easily and efficiently determine the sides and angles of a triangle.

## How to Use the Law of Sines Calculator?

### Input

If you have two angles and one side (AAS or ASA), input these values into the corresponding fields. If you have two sides and one angle that is not between them (SSA), provide these details.

### Calculation

Once you've input all the data you have, click the "Calculate" button.

### Result

The calculator will instantly display the missing side lengths or angles using the law of sines formula.

## What Is the Law of Sines?

The Law of Sines is a fundamental equation in geometry that describes the relationship between the angles and the side lengths of a triangle.

**Formulation**

The Law of Sines states that, in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This means that, for a triangle with sides of lengths $$$a$$$, $$$b$$$, and $$$c$$$ and angles $$$A$$$, $$$B$$$, and $$$C$$$ opposite these sides, the formula can be expressed as

$$\frac{a}{\sin\left(A\right)}=\frac{b}{\sin\left(B\right)}=\frac{c}{\sin\left(C\right)}$$**Proof**

Consider a triangle with a height $$$h$$$ dropped from one vertex, forming two right triangles. Suppose that the height is dropped from $$$B$$$.

Using basic trigonometry, let's find the height from the first triangle using the definition of sine:

$$\sin\left(A\right)=\frac{h}{c}$$From which

$$h=c\sin\left(A\right)$$Similarly, for the second triangle:

$$h=a\sin\left(C\right)$$Since both left-hand sides are equal, equate the right-hand sides:

$$c\sin\left(A\right)=a\sin\left(C\right)$$After rearranging, we have

$$\frac{a}{\sin\left(A\right)}=\frac{c}{\sin\left(C\right)}$$Similarly, we can prove another part of the law of sines by dropping a height from another vertex.

**Example**

Consider a triangle with $$$a=7$$$ units, $$$A=45^0$$$, and $$$B=60^0$$$.

To find the side $$$b$$$, we can set up the following equation using the Law of Sines:

$$\frac{7}{\sin\left(45^0\right)}=\frac{b}{\sin\left(60^0\right)}$$By cross-multiplying and solving for $$$b$$$, we get that

$$b=\frac{7}{\sin\left(45^0\right)}\sin\left(60^0\right)=\frac{7}{\frac{\sqrt{2}}{2}}\frac{\sqrt{3}}{2}=7\sqrt{\frac{3}{2}}$$## What Are the Areas of Application of the Law of Sines?

The Law of Sines is a math rule used in many areas:

**Physics:**The formula is important in vector resolution, especially when analyzing forces that act in non-perpendicular directions.**Astronomy:**Astronomers use the law of sines to determine the positions and distances of celestial bodies based on observations.**Surveying:**Surveyors use the Law of Sines to indirectly measure distances and calculate angles between specific points on a piece of land.**Geography:**It is used in geographical studies, especially in topographical assessments and cartography.

The Law of Sines is indispensable in areas where understanding triangles and the relationship between angles and sides is key.

## Why Choose Our Law of Sines Calculator?

### Precision

Our calculator is carefully designed to provide accurate results every time.

### User-Friendly Interface

With an intuitive layout, users of all levels can easily navigate and get results.

### Fast Results

The calculator performs computations in no time.

### Versatility

Whether you are a student, teacher, engineer, or hobbyist, our calculator is built to meet a variety of needs.

### FAQ

#### What is the Law of Sines?

The Law of Sines states that, in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

#### How do I use the Law of Sines Calculator?

Simply input the known values (sides and angles) of your triangle, and the calculator will determine the unknown sides and/or angles for you.

#### How does the Law of Sines differ from the Law of Cosines?

While both are equations involving trigonometric functions and both are used to solve triangles, the Law of Sines relates sides to their opposite angles, while the Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.