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** **The law of sines is the relationship between the sides and angles of oblique (non-right) triangles. It identifies an oblique triangle's unknown angle or any triangle that isn't a right triangle. Therefore, the sine law should work with the two angles and their side measurements.

In simple terms, the law of sines is the ratio of the length of a triangle’s side to the sine of the angle opposite side, similar to all sides and angles in a given triangle. For instance, if ΔABC is an oblique triangle with sides a, b, and c, then,

The law of sines can be used when you know two angles and one of the triangles or two sides and angles opposite one of them. In some ambiguous cases, the triangle cannot be determined by the data, and the method is likely to give two possible values for the enclosed angle.

The law of sines is a law in trigonometry which gives the ratio of each side of a plan triangle to the sine of the opposite angle. It is the same for all three sides and angles.

The sines law can also be defined as the ratio of the sine of each arc of a spherical triangle to the sine of the opposite angle, similar to three angles and arcs. So it can be given as,

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

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

The law of sines formula connects the lengths of the sides of a triangle to the sines of consecutive angles. The formula a/sinA = b/sinB = c/sinC is used for all triangles other than SSS and SAS triangle.

In the given formula, a, b, and c are the triangle lengths and A, B, and C are the angles of the triangles and can be represented in three different forms:

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

A = 10 units c = 15 units and Angle C = 42º. Find the angle A of the triangle.

**Solution:**

Based on the given data and the sine law formula: a/sinA = b/sinB = c/sinC

⇒ 10/sin A = 15/sin 42º

⇒ sin A/10 = sin 42º/15

⇒ sin A = (sin 42º/15) × 10

⇒ sin A = (sin 42º/15) × 10

⇒ sin A = (0.4885/3) × 2

⇒ sin A = 0.3256

⇒ A = sin^{-1}(0.3256)

⇒ A = 28.36º

**Answer: ****∠****A = 28.36º**

Triangulation is the technique for using the law of sines to compute the remaining sides of a triangle, given two angles and aside. However, in ambiguous cases, you cannot quite determine the triangle by the given data. Hence, to prove the law of sines, let’s look at two oblique triangles:

As given in the first (yellow) triangle:

h/b = sinA

⇒ h = b sinA

And in the second (blue) triangle:

h/a = sinB

⇒ h = a sinB

Also, sin(180º - B) = sinB

Equalizing the h values, you get:

a sinB = b sinA

⇒ a/sinA = b/sinB

Similarly, by deriving a relation between sin A and sin C, you get:

asinC = csinA

⇒ a/sinA = c/sinC

Combine both expressions to get the following sine law:

a/sinA = b/sinB = c/sinC

In trigonometry ratios, sine, cosine, and tangent are commonly used to find the sides of a right triangle and the unknown angles. The law of sines can be applied:

- when two angles and one side is given, and you have to compute the other sides of a triangle,
*or* - when one non-included angle and two sides are given.

According to the sine law, if ∠A, ∠B, and ∠C are the angles between the sides and a, b and c are the lengths of three sides of a triangle, then:

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

But in case you know the value of two angles and the value of one of the sides like:

a = 7 cm, ∠A = 60°, ∠B = 45°, and you need to find b, then using the sine law we know;

a/sin A = b/sin B, we get;

7/sin 60° = b/sin 45°

7/(√3/2) = b/(1/√2)

14/√3 = √2 b

b = 14/(√3√2) = 14/√6

When you apply the sines law to solve a triangle, you may come across two possible solutions, especially when two different triangles are created using the given details. Let's now look at an ambiguous case of solving a triangle using the law of sines.

**Example: **If the side lengths of △ABC are a = 18 and b = 20 with ∠A opposite to 'a' measuring 36º, calculate the measure of ∠B opposite to 'b'?

**Solution:**

Using the sine law, we have sinA/a = sinB/b = sin36º/18 = sin B/20.

⇒ sin B = (8/10) sin36º or B ≈ 28.159º.

However, please note that sin x = sin(180º - x). ∵ A + B < 180º and A + (180º - B) < 180º, while another approximate measurement is 180º - 28.159º = 151.851º.

Ans. The law of sines states the ratio of the side length of a triangle to the sine of the opposite angle, similar on all three sides.

Ans. The law of sines is the ratio of the sides and angles of a triangle, which is explained by the expression** **a/sinA = b/sinB = c/sinC. So here, A, B, and C are the angles of the triangle, and a, b, c are the length of the sides of the triangle.

Ans. Using the triangulation technique, you can use the law of sines to calculate the remaining sides of a triangle when a side and two angles are given. In simple terms, the sine law can be used when two sides and one of the non-enclosed angles are known.

Ans. To use the law of sines, you must know the size of one angle and the length of its opposite side. In the case of a third measurement, it should be on another side or at an angle.

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

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