Law of sines defines the ratio of sides of a triangle and their respective sine angles are equivalent to each other. The other names of the law of sines are sine law, sine rule and sine formula.
The law of sine is used to find the unknown angle or the side of an oblique triangle. The oblique triangle is defined as any triangle, which is not a right triangle. The law of sine should work with at least two angles and its respective side measurements at a time.
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.
The law of sine is explained in detail as follow:
In a triangle, side “a” divided by the sine of angle A is equal to the side “b” divided by the sine of angle B is equal to the side “c” divided by the sine of angle C.
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.
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,
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.
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.
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:
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°
By the sine law, we know;
Now putting the values, we get;
7/sin 60° = b/sin 45°
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.
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:
Both the law of sines and cosines are used to find the unknown angle or an unknown side of a triangle. Let us counter the difference between the two laws.
Law of sines | Law of cosines |
It is used when we are given with: |
a 2 = b 2 + c 2 − 2bc cos A
b 2 = a 2 + c 2 − 2ac cos B
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
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.
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)
The law of sines is used to determine the unknown side of a triangle when two angles and sides are given.
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.
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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