Basic trigonometric identities

                                                        

            TRIGONOMETRIC  IDENTITIES

      

Reciprocal trigonometric identities.         

        

        1.       Sin θ = 1/Cosec θ

                             

                               or

                  

                     Cosec θ = 1/Sin θ

                                                 

         2.        Cos θ = 1/Sec θ 

                       

                             or 

                   Sec θ = 1/Cos θ

                      

               

           3.    Tan θ = 1/Cot θ

                     or

 

                Cot θ = 1/Tan θ

  Ratio Trigonometric Identities

    

            1.     Tan θ = Sin θ/Cos θ

           2.     Cot θ = Cos θ/Sin θ

             

  pythagorean  Identities  (Important)

            1.      sin2 a + cos2 a = 1

              

           2.     1+tan2 a  = sec2 a

           3.     cosec2 a = 1 + cot2 a

 Trigonometric Functions in Terms of  Their Complement

      

          1.      Tan θ = Sin θ/Cos θ

          2.      Cot θ = Cos θ/Sin θ

  

         3        Sin (90 – θ) = Cos θ

       4.       Cos (90 – θ) = Sin θ

       

         5.      Tan (90 – θ) = Cot θ

       6.       Cot ( 90 – θ) = Tan θ

 

         7.       Sec (90 – θ) = Csc θ

        

        8.      Cosec (90 – θ) = Sec θ

   

Trigonometric Identities In Terms of  Supplementary Angles

       

         1.      sin (180°- θ) = sinθ

         2.      cos (180°- θ) = -cos θ

   

         3.      cosec (180°- θ) = cosec θ

     

         4.      sec (180°- θ)= -sec θ

 

        5.       tan (180°- θ) = -tan θ

        6.       cot (180°- θ) = -cot θ

Trigonometric Identities of Opposite Angles  

        1.       Sin (-θ) = – Sin θ

        2.      Cos (-θ) = Cos θ

        3.      Tan (-θ) = – Tan θ

        4.      Cot (-θ) = – Cot θ

        5.      Sec (-θ) = Sec θ

        6.      Cosec (-θ) = -Cosec θ

      

Sum And Difference of Trigonometric Identities

        1.       sin(α+β)=sin(α).cos(β)+cos(α).sin(β)

       2.        sin(α–β)=sinα.cosβ–cosα.sinβ

       3 .       cos(α+β)=cosα.cosβ–sinα.sinβ

       4.        cos(α–β)=cosα.cosβ+sinα.sinβ

               

               

       5.          tan(α+β)=1-tan(α)tan(β)tan(α)+tan(β)​
  

$$tan⁡(�−�)=tan⁡(�)−tan⁡(�)1+tan⁡(�)tan⁡(�)

 

Double Angle Trigonometric Identities  

 

          1.         sin 2θ = 2 sinθ cosθ

         

           2.        cos 2θ = cos2θ – sin2 θ 

                  

                                 = 2 cos2θ – 1 

                                 = 1 – 2sin2 θ

           3.         tan 2θ = (2tanθ)/(1 – tan2θ)

       

  Half  Angle Identities

         1.           sin (θ/2) = ±√[(1 – cosθ)/2]

        2.          cos (θ/2) = ±√(1 + cosθ)/2

 

        3.          tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

Product Sum Trigonometric Identities

       

      1.         Sin A + Sin B = 2 Sin(A+B)/2 . Cos(A-B)/2

      2.         Cos A + Cos B = 2 Cos(A+B)/2 . Cos(A-B)/2

      3.         Sin A – Sin B = 2 Cos(A+B)/2 . Sin(A-B)/2

      4.        Cos A – Cos B = -2 Sin(A+B)/2 . Sin(A-B)/2

Products of Trigonometric Identities

       1.       Sin A. Sin B = [Cos (A – B) – Cos (A + B)]/2

        2.      Sin A. Cos B = [Sin (A + B) + Sin (A – B)] /2

        3.      Cos A. Cos B = [Cos (A + B) + Cos (A – B)] /2

SUMMARY

Trigonometry, is the branch of Mathematics concerned with specific functions 

of angles and their application to calculations. This concept is given by the Greek 

mathematician Hipparchus. There are six functions of an angle commonly used in 

trigonometry. 

Their names abbreviations are sine (sin), cosine(cos), tangent (tan), cotangent (cot), secant (sec),

and cosecant (csc). 

                                                  

Important trigonometric functions

The six important trigonometric functions (trigonometric ratios) are calculated using the 

below formulas and considering the above figure. It is necessary to get knowledge about 

the sides of the right triangle because it defines the set of important trigonometric functions.

                                                   

                                                      

Sine Function:	sin(θ) = Opposite / Hypotenuse
Cosine Function:	cos(θ) = Adjacent / Hypotenuse
Tangent Function:	tan(θ) = Opposite / Adjacent

                                                        

                                Cosec Function:

                                                                   Cosec( θ )      =  Adjacent / Hypotenuse

                                 Sec Function  :           

                                                                     Sec (θ )       =   Opposite / Hypotenuse

                                 Cotangent Function :   

                                                                    Cotangent ( θ)     = Adjacent / opposite

       

Values of Trigonometric functions

              

Trigonometry in Real life

 Trigonometry simply means calculations with triangles. It is a study in mathematics that involves the lengths, heights, and angles of different triangles. The field emerged during the 2nd century , from applications of geometry to astronomical studies, apart from mathematics, trigonometry has applications in the field of physics. If a student is able to grasp the various concepts of trigonometry in school, they are likely to score better in exams. 

Trigonometry formulas have applications in various fields such as construction, design, and other branches of engineering. It is even applied to crime scene investigations. In this article, we have come up with detailed information on different real-life applications of Trigonometry in various fields of our life.

Trigonometry may not have its direct applications in solving practical issues, but it is used in various things that we enjoy so much

Trigonometry can be used to measure the height of a building or mountains:

 In trigonometry ,If you know the distance from where you observe the building and the angle of elevation you can easily find the height of the building. Similarly, if you have the value of one side and the angle of depression from the top of the building you can find and another side in the triangle, all you need to know is one side and angle of the triangle.