Applications of Trigonometry:
A triangle has six important elements; three angles and three sides. In a triangle ABC , The measure of the three angles are usually denoted by 𝛂, 𝛃 , 𝝷 and 𝜸.
The measure of the three sides opposite to them are denoted by a , b , c respectively. If any three out of these sixes elements, out of which at least one side, are given the remaining three elements can be determined. This process of finding the unknown sides or elements is called Solution of the triangle. calculator or table is used for finding the measure of the angle when value of the Trigonometric ratios are given i.e., to find 𝝷 when sin𝝷 = x
Table of Trigonometric Ratios
Mathematicians have constructed the table giving the values of the Trigonometric ratios of large number of angle between 0° and 90°. These are called natural Sine, cosine, tangent etc.
As sine , sec , and tan go on increasing as theta increases from 0° to 90° , so the number in the columns of the differences for Sine , sec, tan are added. Since cos , cosec , and cot decreases from 0° to 90° therefore, fir cos , cosec , cot the numbers in the columns of the differences are subtracted.
Allied angles in Trigonometry
The Angles associated with the basic angles of measures theta to a right angle or multiple are called allied angles. So, the Angles of Measure 90°± 𝝷 , 180± 𝝷 , 270± 𝝷 , 360± 𝝷 , are known as the allied angles.
Fundamental law of Trigonometry
Let 𝛂 and 𝛃any two angles or real numbers ,
Then,
cos (𝛂- 𝛃) = cos𝛂 cos𝛃 + sin𝛂 sin𝛃
Solution of Right angled Triangle
In order to solve a right angle Triangle, we have to find:
- The measure of two acute angles (less than 90°)
- The length of three sides of the triangle.
A Trigonometric ratio of an acute angle of a right triangle involves 3 quantities "Length of two sides and measure of one angle". Thus if two of these three quantities are known , we can find third quantity.
- When measure of two sides are given , we use formula of Cos that is b/c.
- When measure of one side and one angle are given we use the formula of tan that is a/b.
Law of sines
Law of sines is:
- a/sin𝛂 = b/sin𝛃 = c/sin𝜸
Law of cosine
In any triangle ABC , with usual notation.
laws of cosine are:
- a² = b² + c² - 2bc cos𝛂
- b² = c² +a² - 2ac cos𝛃
- c² = a² + b² - 2ab cos𝜸
Laws of tangent
Laws of tangent are:
- a - b/a + b=tan (𝛂 - 𝛃/2)/tan (𝛂 + 𝛃/2)
- b - c/b + c= tan (𝛃 - 𝜸/2)/ tan (𝛃 + 𝜸/2)
- c - a/c + a= tan ( 𝜸 - 𝛂/2)/tan (𝜸 + 𝛂/2)
Half angle formulas
Sine of half the Angles in the terms of sides in any ABC triangle are:
- sin𝛂/2 = √(s - b)(s - c)/bc
- sin𝛃/2 = √(s - a)(s - c)/ac
- sin𝜸/2 = √(s - a)(s - b)/ab
The cosine of the angle in the terms of sides in any ABC triangle are:
- cos𝛂/2 = √s(s - a)/bc
- cos𝛃/2 = √s(s - b)/ac
- cos𝜸/2 = √s(s - c)/ab
The tangent of half the angle in the terms of sides in any ABC triangle are:
- tan𝛂/2 = √(s - b)(s - c)/s(s - a)
- tan𝛃/2 = √(s - a)(s - c)/s(s - b)
- tan𝜸/2 = √(s - a)(s - b)/s(s - c)
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