what is an equation | what is trigonometric equation | techniques of solving trigonometric equations | Knowledge of Physics

-

Solving a mathematical problem is not an easy task. You need to have a lot of background knowledge to solve problems. If you a student of Physics, you need to solve many types of problem using mathematical relations. To be strong in Physics, you need to make yourself strong mathematics as well.  

In this post, you will learn about the following topics:


Learn more about Trigonometric ratios and their formulae from HERE.


1. What is an Equation ?

The mathematical relation of the form ax + bx + c = 0 which is true for certain values of given variable x, is called equation.

For example, 2x+2 = 1 is an equation, which true only for

→ 2x+2 =1

or, 2x = 1-2

or, 2x = - 1

∴ x = \frac{-1}{2}

Thus given equation is true only for x = \frac{-1}{2}. This means calculated value of x satisfies the given equation. 

Another form of equation is quadratic equation of the form

ax^2 + bx +c = 0

You might have leant it in School level. 

Solution or the value of x for this equation is given as

x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a} ---------(i)

For example,

Let the quadratic equation be,

x^2 - 3x +2 = 0 --------(ii)

If given quadratic equation is very hard to factorize, then you can use formula (i) to find the value of variable x. But, if it is easier to factorize, then can factorize it as follows:

\rightarrow x^2 -3x +2 = 0

or, x^2 - 2x -x + 2 = 0

or, x(x-2) - 1(x-2) = 0

or, (x - 2)(x - 1) = 0

Either, x - 2 = 0

OR, x - 1 = 0

\therefore x= 1, 2.

Thus x = 1, 2, which are the required solution of given quadratic equation (ii). You can get same soution by using the formula (i). For which a = 1, b = -3, c = 2. Try it !


2. Trigonometric Equation

The equation that contains trigonometric ratios such as sin2\theta - cos\theta = 0 as variable and is true for certain values of angle (\theta) given in the equation is called trigonometric equation.  The values of trigonometric ratios repeat on certain intervals for the increase or decrease of angle. This is why trigonometric ratios are also called periodic ratios.

Solving given equation:

sin2\theta - cos\theta = 0

or, sin2\theta = cos\theta

or, sin2\theta = sin(90^o \pm \theta)

Both '+' and '-' signs are taken on right hand side. It is because, by Quadrant rule, we know that sine is positive on both of first(0^o\le\theta\le90^o) and second(90^o\le\theta\le180^o)  quadrants.

Either, sin2\theta = sin(90^o + \theta) ------(i)

Or, sin2\theta = sin(90^o - \theta) ------(ii)

From (i),
sin2\theta = sin(90^o + \theta)

or, 2\theta = 90^o + \theta

or, 2\theta - \theta = 90^o

\therefore \theta = 90^o.

From (ii)

sin2\theta = sin(90^o - \theta)

or, 2\theta = 90^o - \theta

or, 3\theta = 90^o

or, \theta = \frac{90^o}{3} = 30^o

\therefore \theta = 30^o.

Hence, the solutions for given trigonometric equation are 90^0 and 30^o.


For example:

If sin\theta = \frac{\sqrt{3}}{2}, then this equation is true only for \theta = 60^o and 120^o. It is because sine of other values except 60^o and  120^o, can not generate \frac{\sqrt{3}}{2}.


Note: Keep in Mind

What is different between an identity relation and an equation ?

An identity relation is a mathematical equation which is true for all values of the variable involved in the relation. However, an equation is true only for certain values of variable involved in the equation.

For example:

Consider two equations:

sin^2\theta + cos^2\theta  = 1 ----(a) and

2sin\theta - \sqrt{3} = 0 ------(b)

Let the values of  angle \theta be 0^o, 30^o, 45^o, 60^0, 90^o, 120^o and 180^o, then

In equation (a),

For \theta = 0^o, 

sin^{2}0 + cos^{2}0 = 1

or, 0 + 1 = 1

\therefore 1 =1, which is true.

For \theta = 30^o,

sin^{2}30^o + cos^{2}30^o = 1

or, (\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2 = 1 

or, \frac{1}{4} + \frac{3}{4} = 1

or, \frac{1 + 3}{4} = 1

or, \frac{4}{4} = 1

\therefore 1 = 1, which is again true.

Similarly, equation (a) is true for all given values of \theta. Check Yourself once 😊 ! It is also true for all other angles, such as  1^o, 2^o, 3^o, 4^o and so on. So, equation (a) is an identity.

But for equation (b), 

let \theta = 0^o,

2 sin0^o - \sqrt{3} = 0

or, 2 \times 0 - \sqrt{3} = 0

or, 0 - \sqrt{3} = 0

\therefore \sqrt{3} = 0, which is not true.

Again, let \theta = 60^0,

2 sin60^o - \sqrt{3} = 0

or, 2 \frac{\sqrt{3}}{2} - \sqrt{3} = 0

or, \sqrt{3} - \sqrt{3} = 0

\therefore 0 = 0, which is true. 

Thus equation (b) is true for only 60^o of given angles. It is not true for other values of  \theta. Thus, equation (b) is just an equation, not an identity. Thus, all identities can be equations, but all equations can't be identities. Keep this difference of identity and equation in your mind !



3. Techniques/Steps of Solving Trigonometric Equation.

Suppose, given trigonometric equation, 0 \leq \theta \leq 360^0 be 

2sin^2\theta + 3 cos\theta  = 3 ------(1)

Now, you have to follow  the following steps to solve trigonometric equation (1).


a)  Bring all terms to Left Hand Side (LHS) making Right Hand Side (RHS) zero. That is,

2sin^2\theta + 3cos\theta = 3

\therefore 2 sin^2\theta + 3cos\theta - 3 = 0 --------(2)


b) Factorize the expression of LHS of equation from (2) making a single factor containing single trigonometric ratio.  That is,

2sin^2\theta + 3cos\theta - 3 = 0

or, 2 (1 - cos^2\theta) +3cos\theta - 3 = 0

or, 2 - 2cos^2\theta + 3cos\theta - 3 = 0

or,  -2cos^2\theta + 3cos\theta -1 = 0

or, 2cos^\theta - 3cos\theta + 1 = 0

or, 2cos^2\theta - 2 cos\theta - cos\theta + 1 = 0

or, 2 cos\theta (cos\theta - 1) -1 (cos\theta - 1) = 0

\therefore (2 cos\theta - 1)(cos\theta - 1) = 0 ------(3)


 c)  LHS on equation (3) contains two factors (2cos\theta - 1) and (cos\theta - 1). Equate each factor to zero. That is

Either, 2 cos\theta - 1 = 0 --------(4)

OR, cos\theta - 1 = 0 --------(5) 


d) Rules of Finding values of angles (\theta):

    i) At first, identify the quadrant on which the least positive value of  angle (\theta) falls. For this, you have to use 'AS The Choice' or 'ASTC' rule. 





Fig. 1 Quadrant Rule for value of Trigonometric Ratios 



ii)  Find the first least positive value of angle of trigonometric ratios in quadrant I, for given relation.



Table.1 Quadrant Rule to determine value of angle for given trigonometric relations.


That is, from (4) and (5),  you have

2cos\theta -1 = 0

\rightarrow cos\theta = \frac{1}{2}




Table.2 Trigonometric Ratios and Their Values for different values of angle(\theta).


As all trigonometric ratios are positive in quadrant I, Then using Table.2, you get

 \theta = cos^{-1}(\frac{1}{2}) = 60^o.

Also,

cos\theta - 1 = 0

\rightarrow cos\theta = 1

\rightarrow \theta = cos^{-1}(1) = 0^o.  

These are the solutions in first quadrant. 


iii) After \theta is found in quadrant I, find other values in remaining quadrants using quadrant rule as

\theta in Quadrant II = 180^o - \theta

\theta in Quadrant III = 180^o + \theta

\theta in Quadrant IV =  160^o - \theta

Find angle more than 360^o just using 360^o + \theta and so on.

Now, 

cos\theta is positive in Quadrant I and IV. Since value of  cos\theta in both equations (4) and (5) are positive So, you need to find values of \theta here  in only quadrant I and IV.

Since, cos\theta = \frac{1}{2},  cos\theta = 1  and values of \theta in quadrant I are 60^o and 0^o respectively. Then 

angle in quadrant IV = 360^o - \theta = 360^o - 60^o = 300^o

and angle in quadrant IV = 260^o - 0^o = 360^o.

\therefore \theta = 0^o, 60^o, 300^o and 360^o.

Hence, the solution for equation (1) are 0^o, 60^o, 300^o and 360^o for the range 0^o \leq \theta \leq 360^o.

This is how you can solve trigonometric equations. 


4. Problems and Solution of some Important Trigonometric Equations


Problem 1: Solve the equation for 0^o \leq \theta 360^o.

sec\theta . tan\theta = \sqrt{2}.

Solution:

Given trigonometric equation is

sec\theta . tan\theta = \sqrt{2} -----(a)

or, \frac{1}{cos\theta} . \frac{sin\theta}{cos\theta} = \sqrt{2}

or, sin\theta = \sqrt{2} cos^2\theta = 0

or, sin\theta - \sqrt{2} cos^2\theta = 0

or, sin\theta - \sqrt{2} (1 - sin^2\theta) = 0

or, sin\theta - \sqrt{2} + \sqrt{2} sin^2\theta = 0

or, \sqrt{2} sin^2\theta + sin\theta - \sqrt{2} = 0

or, \sqrt{2} sin^2\theta  + 2 sin\theta - sin\theta - \sqrt{2} = 0

or, \sqrt{2} sin\theta (sin\theta - \sqrt{2})  - 1 (sin\theta + \sqrt{2}) = 0

or, (\sqrt{2} sin\theta - 1)(sin\theta + \sqrt{2}) = 0

Either, \sqrt{2} sin\theta - 1 = 0 ----(b)

Or, sin\theta + \sqrt{2} = 0 ------(c)

From (b),

\sqrt{2} sin\theta -1 = 0

or, sin\theta = \frac{1}{\sqrt{2}}

or, \theta = sin^{-1}(\frac{1}{\sqrt{2}}) = 45^0 in first quadrant.

Since, sin\theta in (b) has positive value \frac{+ 1}{\sqrt{2}}, then by using quadrant rule of determination of angle, sin\theta is also valid in second quadrant.

Thus, angle in second quadrant = 180^o - \theta = 180^o - 45^0 = 135^o.

Also, from (c),

sin\theta + \sqrt{2} = 0

or, sin\theta = - \sqrt{2}

Since, \sqrt{2} \gt 1, then - \sqrt{2} \lt -1.

But -1 \leq sin\theta \leq 1.

Therefore, (c) is not valid for the solution.

Hence, required solution of given trigonometric equation (a) are 45^o and 135^o.


Problem 2: Solve the following equation for 0^o \leq \theta \leq 180^o.

sin2\theta + cos\theta = 0.

Solution:

Given trigonometric equation is

sin2\theta + cos\theta = 0 -----(a)

or, 2 sin\theta . cos\theta + cos\theta = 0

or, cos\theta (2 sin\theta + 1) = 0

Either, cos\theta = 0 -----(b)

Or, 2 sin\theta + 1 = 0 -----(c)


From (b)

cos\theta = 0

\rightarrow \theta = cos^{-1}(0) = 90^o.

Also from (c),

2 sin\theta + 1 = 0

\rightarrow sin\theta = \frac{-1}{2}

Here sin\theta has negative value.  

However, \theta = sin^{-1}(\frac{1}{2}) = 30^o.

sin\theta is negative only in quadrant III and IV. But by given condition, 0^o \leq \theta 180^o, which allows us to find \theta only in quadrants I and II. So, it is not necessary to find \theta in this case.

Hence, the required solution of equation (a) is 90^o


Jump to Top

Comments

Post a Comment

These posts may also be useful for you

top 3 websites for downloading research papers for free

-
  This post provides you valid links of top 3 websites from where you can download research papers published on high ranked journals for free. Just copy the Title or DOI of the papers, then go to either of these websites and paste on search bar and then hit search button.  1. Sci -hub  2. Library genesis 3.Core Download the file to get the links of these websites. Get File Doing research works is not only learning, it's innovation and interpretation of field of interest.  Watch video 👇👇 Watch Now Don't forget to Follow this blog.  Follow this Blog  
Read More »

plot graph in gnuplot from csv and data file | knowledge of physics

-
 If you are using LINUX based OS, for example Ubuntu, you must be familiar with plotting data in gnuplot saved in a file with .dat extension. It is very simple technique.  To plot data in .dat format, you simply go to gnuplot and run following command, gnuplot> plot 'file_name.dat' using 1:2  This command gives you simple plot of your data saved in  file_name.dat. You can further modify this plot by adding title, axes names, legends and colors of plot and width etc. Here 1:2 means first column of data is along x-axis and second column along y-axis. But what about if you have data in a file of  .csv format ? Let's discuss. CSV means coma separated variables. This means each column of the data in the .csv file is separated by coma (,), and gnuplot does not accept this format directly just as you do for the .dat file.  For gnuplot to read data from .csv file, you should tell it that data you are applying are in coma separated format. So, write code as below...
Read More »

projectile motion in fortran | relation of angle of projection and horizontal range

-
Image
 Have you ever tried to visualize projectile motion in fortran program? This post will help you to generate data and plot the graph of projectile motion.   Above figure shows you the angle of projection(°) versus horizontal range (meter) plot, where the speed of projection is 30 m/s. 👉 Click on link for NEB grade 12  physics numerical problem-solutions:-  LINK Let's start with some basic things: What is projectile and projectile motion ? Projectile:  A projectile is an object thrown to the sky with certain velocity against the earth's gravity. The motion that the projectile possesses during it's journey until it returns to the same surface, from where it was thrown, is called the projectile motion. In above figure, the projectile starts it's journey from point o and follows the curved path to reach another point on the same surface.  Suppose, \theta = angle of projection \therefore Angle of projection is the angle between the horizontal surface of ...
Read More »

NEB physics exam numerical problems with solutions for grade 12 students

-
Image
 This post provides you the numerical problems and their solutions of physics based on NEB(National Examination Board) syllabus grade 12. Solutions of Previously Asked Numerical Problems in NEB exams. Q.1. A coil consisting of 100 circular loops with radius 0.60 cm carries a current of 5A. At what distance from the center, along the axis, the magnetic field has magnitude as \frac{1}{8} great as it is at the center? Solution:- Given, number of circular loops = N = 100 radius of the loop, r = 0.60cm = 6.0 \times 10^-3 m. current, I = 5 A Magnetic field at the center of loop = B (suppose) magnetic field at distance x from the center = \frac{1}{8}B   To find, x = ? The magnetic field at distance x from center of the coil is given as, B = \frac{\mu_o I N r^2}{2(x^2  + r^2)^(\frac{3}{2})}. By given in question, magnetic field at x = \frac{1}{8} \times magnetic field at x = 0.       or, `\frac{\mu_o I N r^2}{2(x^2 + r^2)^(\frac{3}{2})} =\frac{1...
Read More »

how to plot 3D and parametric graphs in gnuplot | three dimensional plot in gnuplot software

-
Image
Although being an open-source plotting software, gnuplot gives you almost all feasibilities to plot 2D, 3D, parametric plots and even the gif as well as animations.  Here you can learn to plot three dimensional graphs in gnuplot. Additionally, you also get the idea of parametric plots in gnuplot software. 1. 3D graphs in gnuplot To plot 3D graphs you just need to replace the command 'plot' for 2D graphs by 'splot' for 3D graphs.  Here, splot means surface plot.  Plotting graphs in 3D view allows you to observe it from all possible sides of the graph.  You rotate it upward, downward , left-side as well as to right-side.  Example: gnuplot >  splot sin(x) Then press ENTER key. You get the output as it looks as picture below: Fig. 1 3D graph of sin(x). 2. parametric plots Parametric plots can be created both in 2D and 3D.  For 2D graph, the dummy variable is t, whereas for 3D plot, they are u and v. Example: Image of parametric plot is shown below: Fig....
Read More »

how to create GIF animation in gnuplot | animation using gnuplot software

-
Image
 GNUPLOT software is not only 2D and/or 3D plotting tool, but it also provides flexibility to create animation and GIF animation plots.  GIF means Graphics Interchange Format. It is a repeated display of images from same of different sides. It helps you to view the image clearly so that you don't need to pay much attention to understand it.  This post provides you tools and techniques to create GIF animation in gnuplot. Codes necessary for creating the GIF are given in links. Please check it as well. GIF can also be both 2D and 3D.  You must use 'plot' command to create 2D GIF and 'splot' command to create 3D GIF. 👉 To learn to create plots in gnuplot: Click Here   👉 To learn gnuplot basics for beginners: Click Here Example of 2D GIF: GNUPLOT code for creating above GIF, 👇 gnuplot > set title 'gif image of sine and cosine functions' gnuplot > set xlabel 'x' gnuplot > set ylabel 'sine and cosine' gnuplot > f1(x,t) = sin(x-t) gnu...
Read More »

NEB board exam maths solved problems with proper solutions - available for free | short answered mathematics problems solved group A

-
Image
 This post provides you the problems with solutions of NEB board exam of grade 11 mathematics 2076. Attempt All Questions. Group A  5 \times 3 \times 2 = 30 Get full question paper from Here . 1. a. Find the truth value of the biconditional statements of p and q where p represents '2 is an even number' and q represents '4 is an even number'. Solution:- Biconditional statements imply if one statement is true then another is also true. if one is false, then another statement is also false. So, one statement supports the other in case of biconditional statement.  Here, p: 2 is an even number.          q: 4 is an even number. 4 is the square of 2 and 2 is the square root of 4. 2 is even, then 4 is also even. So, both p and q represent true value (T). p q p ⇔ q T T T Thus truth value of p and q is T, i.e. 2 is an even number implies 4 is also an...
Read More »

NEB grade 11 and 12 maths exam MCQ solved problems | NEB mathematics MCQ solved problems

-
Image
 NEB  Mathematics grade 11 and 12 specification grid solutions Short Solution Problems(Group A) Rewrite the correct option in your answer sheet. 1. Which of the following is a statement? a) The fishes are beautiful. b) Study Mathematics c) x is the capital of country y. d) Water is essential for health. Correct option: (d) It because any sentence is considered as a statement if it is either true or false but not both. Among given options, (a), (b) and (c)  do not declare truth or falsity. But option (d) declares the truth that water is important for health. 2. The  value of: \sqrt{-16} \times \sqrt{-25} is,  (a) -20 (b) -20i  (c) 20i  (d)20 Correct option: (a) Here,  \sqrt{-16} \times \sqrt{-25} = \sqrt{-1 \times 16} \times \sqrt{-1 \times 25}  But i^2 = -1, i = complex number. = \sqrt{i^2 \times 4^2} \times \sqrt{i^2 \times 5^2} = i \times 4 \times i \times 5 = i^2 \times 20 = -20. // Click here for NEB grade 11 maths short so...
Read More »