Sine rule |cosine rule |sine and cosine rule|sine rule and Circumradius|Geometry

Circumcircle and Circumradius | incircle and inradius | Circumradius and inradius formulae derivation | sine rule | cosine rule

Description - Description - In trigonometry, the Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them. Cosine rule is also called law of cosines or Cosine Formula.

Suppose, a, b and c are lengths of the side of a triangle ABC, then;

a2 = b2 + c2 – 2bc cos ∠x
b2 = a2 + c2 – 2ac cos ∠y
c2 = a2 + b2 – 2ab cos ∠z
where ∠x, ∠y and ∠z are the angles between the sides of the triangle.

The cosine rule relates to the lengths of the sides of a triangle with any of its angles being a cosine angle. With the help of this rule, we can calculate the length of the side of a triangle or can find the measure of the angle between the sides.

Cosine rule

What are the Laws of Cosine?
As per the diagram, Cosine rules to find the length of the sides a, b & c of the triangle ABC is given by;

a2 = b2 + c2 – 2bc cos x
b2 = a2 + c2 – 2ac cos y
c2 = a2 + b2 – 2ab cos z
Similarly, to find the angles x, y and z, these formulae can be re-written as :

cos x = (b2 + c2 -a2)/2bc
cos y = (a2 + c2 -b2)/2ac
cos z = (a2 + b2 – c2)/2ab


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.

Law of Sines Definition
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.

Law of Sines
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.





types of radii for triangles:

inradius: This is the radius of a circle inscribed in a triangle.
circumradius: This is the radius of the circle that is circumscribed on a triangle.

Inradius

When a circle is drawn inside a triangle, and the circle touches three tangent points on the sides of the triangle, the distance from the center to these tangent points is called the inradius of the triangle. It is an inscribed circle, and its center is called the incenter.

Circumradius

The circumradius is the distance from the vertices of the triangles to the center of the circumscribed circle. This circle is also known as the circumcircle.

How do you find the radius of a triangle?

The triangle has two types of radii: the circumradius and the inradius. The circumradius is the distance of the circumcenter to any of the vertices of the triangle. The circumcenter is the center of a circle that is drawn on the triangle that touches all three points of the triangle's vertices. The inradius, on the other hand, is the distance from the center of a circle that is inscribed in a triangle, to the tangent point on the side of a triangle that touches the circle.

What is the difference between circumcircle and circumradius?

Imagine a triangle and a circle is drawn such that all the vertices of the triangle touch the circle. This circumscribed circle is called the circumcircle. The circumradius is the distance from the center of this circle to any of the three vertices of the triangle.

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