Trigonometric Formulae? - Tricks:12th Class

    SUPER HEXAGON TECHNIQUE FOR TRIGONOMETRIC FORMULAS

Tricks to remember formulae in Trigonometry

As Trigonometric functions have a lot of applications in different fields,so by using PALM TRICK , the values of  Trigonometric functions for different standard angles can easily be calculated. 

Palm Trick is very easy and useful method to remember all the values for Trigonometric functions.

Now, let us introduce the SUPER HEXAGON FOR TRIGONOMETRIC FORMULAS.

At left side of Hexagon, there is a trigonometric function like sin,tan,sec and their conversions on right hand side of Hexagon, as follows..

1) sin Ө = cos (90-Ө)

2) tan Ө = cot (90-Ө)

3) sec Ө = cosec (90-Ө)

At vertices all important functions.

CLOCKWISE

4) tan Ө = sin Ө/cos Ө

5) sin Ө = cos Ө/cot Ө

6) cos Ө = cot Ө/cosec Ө

7) cot Ө = cosec Ө/sec Ө

8) cosec Ө = sec Ө/tan Ө

9) sec Ө = tan Ө/sin Ө 

ANTICLOCKWISE

10) tan Ө = sec Ө/cosec Ө

11) sin Ө = tan Ө/sec Ө

12) cos Ө = sin Ө/tan Ө 

Observe Direction of Rays

13) tan Ө X cos Ө = sin Ө

14) tan Ө X cosec Ө = sec Ө

15) cos Ө X cosec Ө = cot Ө

16)  sin Ө X cot Ө = cos Ө

Observe the center of hexagon there is "1"

18) sin Ө X cosec Ө = 1

19)  cos Ө  X  sec Ө = 1

20)  tan Ө X cot Ө = 1

Trigonometric Functions and their values- Tricks

Trigonometry

Trigonometry is one of the most important topics in mathematics.

The first type of trigonometric function, which relates an angle to a side ratio, always satisfies the following equation: f(q) = a / b. 

The secondary trigonometric functions are the sine and cosine of an angle. 

The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (cosec), secant (sec), and cotangent (cot).

Trigonometric functions

The above secondary trigonometric functions are sometimes abbreviated sin(θ) and cos(θ), respectively, 

where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ and cos θ.

• The sine of an angle is defined in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (i.e.the hypotenuse).
• The cosine of an angle is also defined in the context of a right triangle, as the ratio of the length of the side that is adjacent to the angle divided by the length of the longest side of the triangle (i.e.the hypotenuse).
• The tangent (tan) of an angle is the ratio of the sine to the cosine. 

Finally, the reciprocal functions secant (sec), cosecant (cosec), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent.These definitions are sometimes referred to as ratio identities.

Impotance of Trigonometry

• Trigonometric functions relate the angles of a triangle to the lengths of its sides.
• Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. 
• Trigonometric function is used in oceanography in calculating the height of tides in oceans.
•  The sine and cosine functions are fundamental to the theory of periodic functions, those that describe the sound and light waves.
• Trigonometric function has its applications in satellite systems.

Tricks to remember formulae/values of functions in Trigonometry

PALM TRICK : As Trigonometric functions have a lot of applications in different fields, the values of  Trigonometric functions for different standard angles play an important role. So, considering the applications of Trigonometric functions, the values for Trigonometric functions should be remembered.

Palm Trick is very easy and useful method to remember all the values for Trigonometric functions.

So, let us see the algorithmic procedure to remember the values and its tabular form.

Let us arrange the values for sin and cos 

for sin go on counting right side fingers and for cosine go on counting left side fingers for target angle position.

Let us find Sin 0⁰

        for finding sin value of first angle, count right side fingers of the angle.

     for example here angle is 0⁰

     So, here fingers on right side of zero degree = 0

     The formula is Square root of no of fingers / 2

∴       Sin 0⁰  = √0/2 = 0       and   cos 0⁰    =    √4/2 = 2/2 = 1

      Sin 30⁰  = √1/2 =1/2    and   cos 30⁰ =    √3/2                         

      Sin 45⁰ = √2/2 = 1/√2 and   cos 45⁰ = √2/2 = 1/√2 

      Sin 60⁰ = √3/2            and   cos 60⁰  =    √1/2 = 1/2 

        Sin 90⁰ = √4/2 = 2/2   and   cos 90⁰  =    √0/2 = 0  

                              = 1   
Arrange the above calculated values in tabular form for 0⁰,30⁰,45⁰,60⁰ ,90⁰,180⁰ ,360⁰       

Engg. Physics I : paper set 2 :Quantum Mechanics, Semiconductors, Ctrystallography,

Paper Set -  II

Subject: Engineering Physics I

Time: 2:00 Hours ]                                                [  Marks: 40 

INSTRUCTIONS TO CANDIDATES          

All Questions carry Equal Marks as indicated.

Solve FOUR questions as instructed

Assume suitable data

Draw neat sketches wherever necessary.

Use of calculator is permitted.

        Q1. (a)Write Plank’s Quantum Hypothesis with energy level diagram        (2)  
         (b)State any four Properties of Photons.                                                         (2)  
      (c)What is Compton Effect? Explain Compton shift with equation.
            Prove it graphically.                                                                                 (3)         
     (d)If the incident radiation is 1.16 Ă. Find the wavelength of scattered radiations at angle 45⁰.                                                                                                             (3)                                                               OR

      Q2. (a)Prove that light exhibit both wave as well as particle nature.              (2)

(    (b)Explain that the wavelength of Macroscopic bodies is insignificant in comparison  to the size of the bodies. Calculate the value for wavelength.     (2)

       (c)Explain this statement with the help of Davisson & Germer  Experiment.(3)

       (d)Find the De-broglie wavelength of an electron accelerated through a potential difference of 168 v                                                                                                 (3)  

    Q3. (a) Calculate the nearest neighboring distance for SC and BCC               (4)

       (b)Find  out APF, Co ordination No. in BCC and FCC                                  (2) 

       (c)Differentiate between Tetrahedral and Octahedral Voids.                         (2) 

      (d)Molybdenum belongs to BCC lattice. Its density is 10.2 X 10³ Kg/m³and its atomic weight is 95.94. Determine the radius of  Molybdenum  atom.           (2)

                                                            OR

     Q4. (a) Deduce a relation between an interlinear distance d and the Miller indices of the planes for cubic crystal.                                                                           (4)

      (b) Draw the following Miller Indices  planes [1 ̅10],[231],[112]                    (3)

      (c) Explain and deduce Bragg’s Law for X-Ray diffraction.                         (3)

     Q5. (a) Define phase velocity and group velocity.                                             (2)

     (b)Explain the application of Heisenberg’s Uncertainty principle with the help of Thought experiment.                                                                                           (4)

    (c) Explain the interpretation of wave function and write the normalization of wave function.                                                                                                     (2)     

     (d)Write 3D time dependent Schrodinger Equation.                                      (2)

                                                           OR

      Q6. (a)Write three dimensional time dependent Schrodinger Equation.    (2)

       (b)Deduce the equation for Wave function of particle confined in an infinite one dimensional potential Well of length ‘L’                                                         (4)    

       (c)Discuss the tunnling effect with the help of schrodinger  Equation.      (2)

     (d)Calculate the value of lowest two energy state for electron confined in the infinite potential well of width 15 Ǻ (m = 9.11 x 10^-31 Kg, h = 6.626 x 10^-34 J.s) (2)

     Q7. (a)Explain p-type and n- type semiconductors with suitable diagram.       (3)

          (b)Write down the Fermi - Dirac Equation for the probability of occupation of  an energy level E by an electron.                                                                            (4)

       (c)Explain how fermi level changes with increasing amounts of impurity in n-type and p-type Semiconductor.                                                                          (3)

                                                             OR

     Q8. (a)What is Hall Effect?                                                                                      (2)

       (b)Give expression for each of the following parameters Hall Voltage, Hall Coefficient, Hall Mobility.                                                                                        (6)

       (c)Mention the importance of  Hall Effect in the field of Semiconductor.         (2)                                                                                                        

Engg. Physics I: paper set 1:Quantum Mechanics, Semiconductors, Ctrystallography ETC

Paper Set -  I

Subject: Engineering Physics I

Time: 2:00 Hours ]                                                                                 [  Marks: 40 

            INSTRUCTIONS TO CANDIDATE

      All Questions carry Equal Marks as indicated.   

      Solve FOUR questions as instructed.

      Assume suitable data.

       Draw neat sketches wherever necessary.

      Use of calculator is permitted.

Q1. (a)Write Plank’s Quantum Hypothesis with energy level diagram                                       (2)

       (b)State any four Properties of Photons.                                                                                (2)

       (c)What is Compton Effect? Explain Compton shift with equation. Prove it graphically    (3)

       (d)If the incident radiation is 1.16 Ă. Find the wavelength of scattered radiations 

           at angle 45⁰.                                                                                                                        (3)

OR

Q2. (a)Prove that light exhibit both wave as well as particle nature.                                           (2)

       (b)Explain that the wavelength of Macroscopic bodies is insignificant in comparison

             to the size of the bodies. Calculate the value for wavelength.                                         (3)

       (c)Explain this statement with the help of Davisson & Germer  Experiment                (3)

       (d)Find the De-broglie wavelength of an electron accelerated through a potential 

            difference of 168 volts.                                                                                                      (2) 

Q3. (a)Define: Crystal Structure,Non primitive Cell, lattice Planes.                                            (3)

       (b)Find  out Atomic Packing Fraction and Density in SC and FCC                                      (2)

       (c)Differentiate between Tetrahedral and Octahedral Voids.                                                 (2)
       (d)Nickel crystallizes in a FCC crystal. The edge of unit cell is 3.52 Å.  
           The its atomic weight of nickel is 58.710 Kg/Kmol. Determine the density of metal.     (2)

OR

Q4. (a) Deduce a relation between an interlinear distance d and the Miller indices

            of the planes for cubic crystal.                                                                                          (4) 
       (b) Draw the following Miller Indices Priciple planes .                                                        (3)

       (c) Explain and deduce Bragg’s Law for X-Ray diffraction.                                                (3)

Q5. (a) Explain how wave packet are formed taking it in the form of beats.                              (3)
      (b)Explain the application of Heisenberg’s Uncertainty principle with the help of 
           macroscopic body.                                                                                                             (3)

       (c)Explain the interpretation of wave function and write the normalization of 

            wave function.                                                                                                                  (2) 
       (d)Write 3D time independent Schrodinger Equation.                                                          (2)

OR

Q6. (a)Write one dimensional time dependent Schrodinger Equation.                                       (2)

       (b)Show that the Wave function for particle confined in an infinite one dimensional
          potential well of length ‘l’ is given by Ψn(x) = √2/l sin(nПx/l)                                         (4)

       (c)Discuss the Tunnling effect with the help of Schrodinger Equation.                                (2)

       (d)Calculate the value of lowest three energy state for electron confined in the infinite

            potential well of width 10 Ǻ (m = 9.11 x 10^-31 Kg, h = 6.626 x 10^-34 J.s)                     (2)

 Q7. (a)Explain p-type and n- type semiconductors with suitable diagram.                                (3)

       (b)Write down the Fermi - Dirac Equation for the probability of occupation of  an 

             energy level E by an electron.                                                                                         (4)

       (c)Explain how fermi level changes with increasing amounts of impurity in n-type

           and p-type Semiconductor.                                                                                                (3)

OR

Q8. (a)What is Hall Effect with suitable experimental diagram?                                                (3)

       (b)Calculate each of the following parameters Hall Voltage, Hall Coefficient,

            Hall Mobility.                                                                                                                   (5)

       (c)Mention the importance of  Hall Effect in the field of Semiconductor.                          (2)