**Math Calculators** ▶ Law of Sines Calculator

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An online law of sines calculator allows you to find the unknown angles and lengths of sides of a triangle. When we dealing with simple and complex trigonometry sin(x) functions, this calculator uses the law of sines formula that helps to find missing sides and angles of a triangle.

So, read on to get a complete guide about sine laws.

The Laws of sines are the relationship between the angles and sides of a triangle which is defined as the ratio of the length of the side of a triangle to the sine of the opposite angle.

Where:

**Sides of Triangle:**

$$a = side a, b = side b, c = side c$$

**Angles of Triangle:**

$$A = angle A, B = angle B, C = angle C$$

**Characteristics of Triangle:**

P = Triangle perimeter, s = semi-perimeter, K = area, r = radius of inscribed circle, R = radius of circumscribed circle

If a, b, and c are the length of sides of a triangle and opposite angles are A, B, and C respectively; then law of sins shows:

$$a/sinA = b/sinB = c/sinC$$

So, the law of sine calculator can be used to find various angles and sides of a triangle.

**Example:**

Compute the length of sides b and cÂ of the triangle shown below.

**Solution:**

Here, calculate the length of the sides, therefore, use the law of sines in the form of

\(\frac{a}{sin A} = \frac{b}{sin B}\)

Now,

$$\frac{a}{sin 100^0}= \frac{12}{sin 50^0}$$

By Cross multiply:

$$12 sin 100^0= a sin 50^0$$

Both sides divide by sin \(50^0\)

$$a = \frac{(12 sin 100^0)}{sin 50^0}$$

From the calculator we get:

$$a = 15.427$$

So, the length of sides b and c is \(15.427 mm\).

However, an Online Sine Calculator will calculate the sine trigonometric function for the given angle in degree, radian, or the Ï€ radians.

These are some equations that are used by the law of sines calculator which are obtained from the law of sins:

$$A=sin^{âˆ’1}[\frac{asinB}{b}]$$

$$A=sin^{âˆ’1}[\frac{asinC}{c}]$$

$$B=sin^{âˆ’1}[\frac{bsinA}{a}]$$

$$B=sin^{âˆ’1}[\frac{bsinC}{c}]$$

$$C=sin^{âˆ’1}[\frac{csinA}{a}]$$

$$C=sin^{âˆ’1}[\frac{csinB}{b}]$$

In order to find any side of a triangle law of sines calculator fetched some values from law of sines formula:

$$a=\frac{bsinA}{sinB}$$

$$a=\frac{csinA}{sinC}$$

$$b=\frac{asinB}{sinA}$$

$$b=\frac{csinB}{sinC}$$

$$c=\frac{asinC}{sinA}$$

$$c=\frac{bsinC}{sinB}$$

Also, you can find alpha (Î±) by using, \(a=n/a, b=n/a, beta (Î² ) =n/a\) values, while the value of beta is calculated by using \(a=n/a, alpha=n/a, b=n/a\).

An ambiguous case occurs, when two different triangles constructed from given data then the triangles are \(ABC \text{ and} ABâ€™Câ€™\).

There are some conditions to use the law of sines for the case to be ambiguous:

- When only sin(a)sin(b) and an angle A given.
- The angle of A is less than \(90^0\).
- Side a is shorter as compared to side c.
- Side a is longer than altitude h from the angle B where a > h.

Furthermore, The online CSC Calculator will determine the cosecant and sin inverse trigonometric function for the given angle it either in degree, radian, or the pi (Ï€) radians.

The law of sine calculator especially used to solve sine law related missing triangle values by following steps:

- You have to choose an option to find any angle or side of a trinagle from the drop-down list, even the calculator display the equation for the selected option
- Now, you need to add the value for angles and sides into the designated fields
- Then, you have to select the units for the entered values
- At last, make a click on the given calculate button

The law of sines calculator calculates:

- The value of angles and sides for the given equation
- The values for the different characteristics ofÂ a triangle
- Diagram

When you have two sides and one angle or two angles and one side of a triangle then we use laws of sines.

According to the triangle inequality theorem, the sum of any two sides must be greater than the third side of a triangle and this rule must fulfil all three conditions of sides.

An oblique triangle is not a right triangle so basic trigonometric ratios do not apply, they can be modified to cover oblique by using sines and cosines law.

There are different ways to find triangle characteristics:

- Radius of circle around triangle \(R = (abc) / (4K)\)
- Radius of inscribed circle in a triangle \(r = \sqrt{(s-a)*(s-b)*(s-c) / s}\)
- Triangle area \(K = \sqrt{ s*(s-a)*(s-b)*(s-c)}\)
- Triangle semi perimeter \(s = 0.5 * (a + b + c)\)
- Perimeter \(P = a + b + c\)

The law of sines calculator is highly recommendable for assessing the missing values of a triangle by using the law of sines formula. Finding all these values manually is a difficult task, also it increases the risk to get accurate results. By using the law of sine calculator you can find all sine law values instantly and 100% error-free. Moreover, this tool is beneficial for people who work with the law of sine related trigonometric function.

From the source of Wikipedia: The ambiguous case of triangle solution, Relation to the circumcircle, Relationship to the area of the triangle.

From the source of Dave’s Short Trig Course: Oblique Triangles, Pythagorean theorem, Triangle Characteristics.

From the source of Khan Academy: Laws of sines and cosines review, Solving triangles using the law of sines, Missing Angle.