Chapter-2-EQUATION-2B-01-Theory

2.1 Introduction . An equation involving one or more trigonometrical ratio of an unknown angle is called a trigonometrical equation i.e., , ; etc. A trigonometric equation is different from a trigonometrical identities. An identity is satisfied for every value of the unknown angle e.g., is true while a trigonometric equation is satisfied for some particular values of the unknown angle. (1) Roots of trigonometrical equation : The value of unknown angle (a variable quantity) which satisfies the given equation is called the root of an equation e.g., , the root is or because the equation is satisfied if we put or . (2) Solution of trigonometrical equations : A value of the unknown angle which satisfies the trigonometrical equation is called its solution. Since all trigonometrical ratios are periodic in nature, generally a trigonometrical equation has more than one solution or an infinite number of solutions. There are basically three types of solutions: (i) Particular solution : A specific value of unknown angle satisfying the equation. (ii) Principal solution : Smallest numerical value of the unknown angle satisfying the equation (Numerically smallest particular solution.) (iii) General solution : Complete set of values of the unknown angle satisfying the equation. It contains all particular solutions as well as principal solutions. When we have two numerically equal smallest unknown angles, preference is given to the positive value in writing the principal solution. e.g., has etc. As its particular solutions out of these, the numerically smallest are and but the principal solution is taken as to write the general solution we notice that the position on P or can be obtained by rotation of OP or OP around O through a complete angle any number of times and in any direction (clockwise or anticlockwise)  The general solution is . 2.2 General Solution of Standard Trigonometrical Equations . (1) General solution of the equation sin  = sin: If or or,  or, or  or  = (any even multiple of ) +  or  = (any odd multiple of ) –  Note :  The equation is equivalent to . So these two equation having the same general solution. (2) General solution of the equation cos = cos : If    or ,  or  or . for the general solution of , combine these two result which gives Note :  The equation is equivalent to so the general solution of these two equations are same. (3) General solution of the equation tan  = tan  : If     Note :  The equation is equivalent to so these two equations having the same general solution. 2.3 General Solution of Some Particular Equations . (1)  ,  or ,  (2)  or ,  ,  or (3)  or ,  ,  or (4) = not defined  , = not defined  = not defined  , = not defined  . Important Tips  For equations involving two multiple angles, use multiple and sub-multiple angle formulas, if necessary.  For equations involving more than two multiple angles (i) Apply formula to combine the two.(ii) Choose such pairs of multiple angle so that after applying the above formulae we get a common factor in the equation. Example: 1 If then the general value of is (a) (b) (c) (d) Solution: (c) Example: 2 The general solution of is (a) (b) (c) (d) Solution: (b) Example: 3 If then the general value of is (a) (b) (c) (d) None of these Solution: (b) or For (m) even i.e., then And for (m) odd, i.e., then Example: 4 The general solution of is (a) (b) (c) (d) Solution: (d) (which is impossible)   . Example: 5 The number of solutions of the equation in the interval is (a) 0 (b) 5 (c) 6 (d) 10 Solution: (c)  , But so . Hence from solution's (one in Ist quadrant and other in 2nd quadrant), from solution's and solution's. So total number of solutions Example: 6 Number of solutions of the equation lying in the interval is (a) 0 (b) 1 (c) 2 (d) 3 Solution: (c) So, or or but does not satisfy the equation, So total number of solutions Example: 7 If then (a) (b) (c) (d) None of these Solution: (c) . Example: 8 The solution of the equation is (a) (b) (c) (d) None of these Solution: (c) Given equation is As therefore . Example: 9 The solution of the equation is (a) (b) (c) (d) Solution: (b) After solving the determinant Example: 10 The general value of in the equation is (a) (b) (c) (d) Solution: (c) (impossible) or Example: 11 The general value of is obtained from the equation is (a) (b) (c) (d) Solution: (d)   Example: 12 If then (a) (b) (c) (d) Solution: (a) On expanding the determinant or or or Example: 13 If then the general value of is (a) (b) (c) (d) Solution: (a)  or Example: 14 then equal to (a) (b) (c) (d) None of these Solution: (a) or or or or . 2.4 General Solution of Square of Trigonometrical Equations . (1) General solution of sin2 = sin2  : If or, (Both the sides multiply by 2) or, or, , , (2) General solution of cos2  = cos2  : If or, (multiply both the side by 2) or, or, ; (3) General solution of tan2  = tan2: If or, Using componendo and dividendo rule, or or or , Example: 15 General value of satisfying the equation is (a) (b) (c) (d) None of these Solution: (b)     and and  and Example: 16 If then the general value of is (a) (b) (c) (d) Solution: (b) . Example: 17 If , then the general value of is (a) (b) (c) (d) Solution: (c)  Example: 18 If , then the most general value of is (a) (b) (c) (d) Solution: (c) Example: 19 If then the general value of is (a) (b) (c) (d) Solution: (d) . 2.5 Solutions in the Case of Two Equations are given (Simultaneously Solving Equation). We may divide the problem into two categories. (1) Two equations in one ‘unknown’ satisfied simultaneously. (2) Two equations in two ‘unknowns’ satisfied simultaneously. (1) Two equations is one ‘unknown’ : Two equations are given and we have to find the values of variables which may satisfy with the given equations. (i) and , so the common solution is (ii) and , so the common solution is (iii) and , so the common solution is Example: 20 The most general value of satisfying the equation and is (a) (b) (c) (d) None of these Solution: (c) and Hence, general value is Example: 21 The most general value of which will satisfy both the equations and is (a) (b) (c) (d) None of these Solution: (d) Hence, general value of is (2) System of equations (Two equations in two unknowns) : Let be the system of two equations in two unknowns. Step (i) : Eliminate any one variable, say Let be one solution. Step (ii) : Then consider the system and use the method of two equations in one variable. Note :  It is preferable to solve the system of equations quadrant wise. Example: 22 If then the value of and are (a) (b) (c) (d) None of these Solution: (a) But, so that Trick: Check with the options for . Example: 23 Solve the system of equations Solution: Usually students proceed this type of problems in the following way: Squaring and subtracting, we get i.e., or or .........(i) Also we have which gives or and so Thus solution of this system is and ........(ii) Now see the fallacies: and (from the solution) give i.e., but give Thus solution given in (ii) consists many extraneous (absurd) solutions. The simple reason for this is quite obvious. (ii) consists of solutions of following four systems: .........(iii) .........(iv) ..........(v) and .........(vi) While we have to find the values which satisfy (iii). Therefore, we have to verify the solutions and should retain only the valid ones. Alternative Method : A better method for such type of equations is following: The given system is ........(vii) ........(viii) (vii)2 – (viii)2 gives Case 1 : the system reduces to so .........(ix) Case 2 : then we have so . Thus general solution is ........(x) Case 3 : (or can be taken as ) Then so Thus or ........(xi) Case 4: (or Then so and so ..........(xii) Hence, the required solutions are given as ; Note :  Do not write the solution as . 2.6 General Solution of the form a cos + bsin = c . In put and where and Then, (say) ..........(i) where is the general solution Alternatively, putting and where (say) where , is the general solution. Note :   The general solution of is . Example: 24 The number of integral values of k, for which the equation has a solution is (a) 4 (b) 8 (c) 10 (d) 12 Solution: (b) So, for solution or or or . So, integral values of k are (eight values) Example: 25 If , then general value of is (a) (b) (c) (d) Solution: (d) . Example: 26 If , then the general value of is (a) (b) (c) (d) None of these Solution: Example: 27 The equation has (a) Only one solution (b) Two solutions (c) Infinitely many solutions (d) No solution Solution: (d) Given equation is which is of the form with Here Therefore the given equation has no solution. Example: 28 The general solution of the equation is (a) (b) (c) (d) Solution: (a) Divided by in both sides, We get,    . 2.7 Some Particular Equations . (1) Equation of the form : Here are real numbers and the sum of the exponents in and in each term is equal to n, are said to be homogeneous with respect to sinx and cosx. For above equation can be written as, Example: 29 The solution of equation is (a) or (b) or (c) or (d) None of these Solution: (a) To solve this kind of equation; we use the fundamental formula trigonometrical identity, writing the equation in the form, Dividing by on both sides we get, Now it can be factorized as; i.e., or or (2) A trigonometric equation of the form : Here R is a rational function of the indicated arguments and (k, l, m, n are natural numbers) can be reduced to a rational equation with respect to the arguments by means of the formulae for trigonometric functions of the sum of angles (in particular, the formulas for double and triple angles) and then reduce equation of the given form to a rational equation with respect to the unknown, by means of the formulas, Example: 30 If then (a) (b) (c) (d) None of these Solution: (a) Let and using the formula. We get, Its roots are; and Thus the solution of the equation reduces to that of two elementary equations, is required solution. (3) Equation of the form : where R is rational function of the arguments in brackets, Put ........(i) and use the following identity: …….(ii) Taking (i) and (ii) into account, we can reduce given equation into; . Similarly, by the substitution we can reduce the equation of the form; to an equation; Example: 31 If then the general solution of x is (a) (b) (c) Both (a) and (b) (d) None of these Solution: (c) Let and using the equation we get The numbers are roots of this quadratic equation. Thus the solution of the given equation reduces to the solution of two trigonometrical equation; or or or or or or or or . Example: 32 If then (a) (b) (c) (d) None of these Solution: (b) Using half-angle formulae we can represent the given equation in the form, Put whose only real root is, ; I Note :  Some trigonometric equations can sometimes be simplified by lowering their degrees. If the exponent of the sines and cosines occuring into an equation are even, the lowering of the degree can be done by half angle formulas as in above example. 2.8 Method for Finding Principal Value . Suppose we have to find the principal value of  satisfying the equation . Since is negative,  will be in 3rd or 4th quadrant. We can approach 3rd or 4th quadrant from two directions. If we take anticlockwise direction the numerical value of the angle will be greater than . If we approach it in clockwise direction the angle will be numerically less than . For principal value, we have to take numerically smallest angle. So for principal value (1) If the angle is in 1st or 2nd quadrant we must select anticlockwise direction and if the angle is in 3rd or 4th quadrant, we must select clockwise direction. (2) Principal value is never numerically greater than . (3) Principal value always lies in the first circle (i.e., in first rotation). On the above criteria,  will be or Among these two has the least numerical value. Hence is the principal value of  satisfying the equation From the above discussion, the method for finding principal value can be summed up as follows : (i) First draw a trigonometrical circle and mark the quadrant, in which the angle may lie. (ii) Select anticlockwise direction for 1st and 2nd quadrants and select clockwise direction for 3rd and 4th quadrants. (iii) Find the angle in the first rotation. (iv) Select the numerically least angle. The angle thus found will be principal value. (v) In case, two angles one with positive sign and the other with negative sign qualify for the numerically least angle, then it is the convention to select the angle with positive sign as principal value. Example: 33 If then (only principal value) (a) (b) (c) (d) Solution: (a) Example: 34 Principal value of is (a) (b) (c) (d) Solution: (a) is negative. will lie in 2nd or 4th quadrant. For 2nd quadrant we will select anticlockwise and for 4th quadrant, we will select clockwise direction. In the first circle two values and are obtained. Among these two, is numerically least angle. Hence principal value is Important Tips  Any trigonometric equation can be solved without using any formula. Find all angles in which satisfy the equation and then add to each. For example: Consider the equation then Hence required solutions are 2.9 Important Points to be Taken in Case of While Solving Trigonometrical Equations . (1) Check the validity of the given equation, e.g., can never be true for any as the value can never exceeds . So there is no solution to this equation. (2) Equation involving or can never have a solution of the form. Similarly, equations involving or can never have a solution of the form . The corresponding functions are undefined at these values of . (3) If while solving an equation we have to square it, then the roots found after squaring must be checked whether they satisfy the original equation or not, e.g., Let . Squaring, we get  and but does not satisfy the original equation . e.g., Square both sides, we get  or ,  Roots are ……, We find that 0 and are roots but and do not satisfy the given equation as it leads to Similarly 0 and are roots but and are not roots as it will lead to . As stated above, because of squaring we are solving the equations and both. The rejected roots are for . (4) Do not cancel common factors involving the unknown angle on L.H.S. and R.H.S. because it may delete some solutions. e.g., In the equation if we cancel on both sides we get . But also satisfies the equation because it makes . So, the complete solution is . (5) Any value of x which makes both R.H.S. and L.H.S. equal will be a root but the value of x for which will not be a solution as it is an indeterminate form. Hence, for those equations which involve and whereas for those which involve and . Also exponential function is always +ve and is defined if , and always and not and not . (6) Denominator terms of the equation if present should never become zero at any stage while solving for any value of  contained in the answer. (7) Sometimes the equation has some limitations also e.g., can be true only if and simultaneously as . Hence the solution is . (8) If then either or or both. But only and not , as it will make . Similarly if , then it will also imply only as being a constant. Similarly and . Here we do not take as in the above because x is an additive factor and not multiplicative factor. When , then or . We have to verify which value of is to be chosen which satisfies the equation. . (9) Student are advised to check whether all the roots obtained by them, satisfy the equation and lie in the domain of the variable of the given equation. 2.10 Miscellaneous Examples . Example: 35 The equation is satisfied if (a) (b) (c) (d) Solution: (b)  on solving, . Either (which is not possible) or . Example: 36 If the solutions for of are in A.P., then the numerically smallest common difference of A.P. is (a) (b) (c) (d) Solution: (b) Given, or , . Both the solutions form an A.P. gives us an A.P. with common difference and gives us an A.P. with common difference = . Certainly, . Example: 37 The set of values of x for which the expression =1 is (a) (b) (c) (d) Solution: (a) But this value does not satisfy the given equation. Example: 38 If then (a) (b) (c) (d) None of these Solution: (c) Combining and we get Hence Example: 39 If , then equal to (a) (b) (c) (d) None of these Solution: (b)   . Example: 40 The sum of all solutions of the equation is (a) (b) (c) (d) None of these Solution: (b) Here, or or where ; The required sum = 30. Example: 41 The equation for has (a) One solution (b) Two sets of solution (c) Four sets of solution (d) No solution Solution: (a) Given, is satisfied only when for . Example: 42 The solution set of in the interval is (a) (b) (c) (d) Solution: (c) which is not possible or Solution set is Example: 43 The equation has (a) Finite solution (b) Infinite solution (c) One solution (d) No solution Solution: (d) So that equation has no solution. Example: 44 The equation has (a) One solution (b) Two solution (c) Infinite number of solution (d) No solution Solution: (d) No solution as and both of them do not attain their maximum value for the same angle. Trick: Maximum value of Hence there is no x satisfying this equation. Example: 45 If and then the solution set for x is (a) (b) (c) (d) None of these Solution: (a) Here, The value scheme for this is shown below. From the figure, or . Example: 46 The number of pairs (x, y) satisfying the equations and is (a) 2 (b) 4 (c) 6 (d) Solution: (c) The first equation can be written as, Either or or  . As |x| + |y| =1, therefore when we have to reject or and solve it with or which gives or as the possible solution. Again solving with we get and solving with we get as the other solution. Thus we have six pairs of solution for x and y. Example: 47 If then the values of are (a) (b) (c) (d) Solution: (c) Given, and We know that and or Similarly or Therefore and 240°. Example: 48 If then the value of (a) (b) (c) (d) Solution: (a) Example: 49 The only value of x for which hold is (a) (b) (c) (d) All values of x. Solution: (a) Since A.M. G.M. And, we know that for 2.11 Periodic Functions . A function f(x) is called periodic function if there exists a least positive real number T such that T is called the period (or fundamental period) of function . Obviously, if T is the period of then (i) If and are two periodic functions of x having the same period T, then the function where a and b are any numbers, is also a periodic function having the same period T. (ii) If T is the period of the periodic function , then the function where and b are any numbers is also a periodic function with period equal to (iii) If and are the periods of periodic functions and respectively, then the function where a and b are any numbers is also periodic and its period is which is the L.C.M. of and i.e. T is the least positive number which is divisible by and All trigonometric functions are periodic. The period of trigonometric function and is because etc. The period of and is because and The period of the function which are of the type: The period of is Here |a| is taken so as the value of the period is positive real number. Some functions with their periods Function Period Example: 50 Period of is (a) (b) (c) (d) None of these Solution: (a) Period Example: 51 The period of the function is (a) (b) (c) (d) Solution: (a) Period of Period of Example: 52 The period of the function is (a) (b) (c) (d) Solution: (d) Period of and period of L.C.M. of and Example: 53 The function is periodic with period (a) 6 (b) 3 (c) 4 (d) 12 Solution: (d) Period of Period of and period of Period of L.C.M. of (4, 6, 4)=12. Example: 54 If the period of the function is then n is equal to (a) 1 (b) 4 (c) 8 (d) 2 Solution: (d) Period of the function is Example: 55 The period of is (a) (b) (c) (d) Solution: (a) Therefore period is Trick:   Hence the period is Example: 56 Period of is (a) (b) (c) (d) Solution: (d) Hence period . Example: 57 Period of is (a) (b) (c) (d) Solution: (b) Period of and period of