Relation between Angles and Sides of Triangle

Three rules which related the sides of triangles to the interior angles of triangles are,

• Sine Rule
• Cosine Rule
• Tangent Rule

If a triangle ABC with sides a, b, and c which are sides opposites to the ∠A, ∠B, and ∠C respectively, then

Sine Rule

Sine rule states the relationship between sides and angles of the triangle which is the ratio of side and sine of angle opposite to the side always remains the same for all the angles and sides of the triangle and is given as follows:

[Tex]\bold{\frac{\sin \angle A}{a}= \frac{\sin \angle B}{b} = \frac{\sin \angle C}{c} = k} [/Tex]

Cosine Rule 

Cosine Rule involves all the sides, and one interior angle of the triangle is given as follows:

[Tex]\bold{\cos \angle A = \frac{b^2+c^2 – a^2}{2bc}} [/Tex]

OR

[Tex]\bold{\cos \angle B = \frac{a^2+c^2 – b^2}{2ac}} [/Tex]

OR

[Tex]\bold{\cos \angle C = \frac{a^2+b^2 – c^2}{2ab}} [/Tex]

 

Tangent Rule

• Tangent Rule also states the relationship between the sides and interior angle of a triangle, using the tan trigonometric ratio, which is as follows:

• [Tex]\bold{\frac{a-b}{a+b}=\frac{\tan \left(\frac{A-B}{2}\right)}{\tan \left(\frac{A+B}{2}\right)}} [/Tex]
• [Tex]\bold{\frac{b-c}{b+c}=\frac{\tan \left(\frac{B-C}{2}\right)}{\tan \left(\frac{B+C}{2}\right)}} [/Tex]
• [Tex]\bold{\frac{c-a}{c+a}=\frac{\tan \left(\frac{C-A}{2}\right)}{\tan \left(\frac{C+A}{2}\right)}} [/Tex]

Also, Read

• Trigonometry Height and Distance
• Trigonometric Table

Trigonometric Identities are various identities that are used to simplify various complex equations involving trigonometric functions. Trigonometry is a branch of Mathematics that deals with the relationship between the sides and angles of a triangle., These relationships are defined in the form of six ratios which are called trigonometric ratios – sin, cos, tan, cot, sec, and cosec.

In an extended way, the study is also of the angles forming the elements of a triangle. Logically, a discussion of the properties of a triangle; solving a triangle, and physical problems in the area of heights and distances using the properties of a triangle – all constitute a part of the study. It also provides a method of solution to trigonometric equations.