TRIGONOMETRIC FUNCTIONS

Transcript

1 Chapter TRIGONOMETRIC FUNCTIONS. Overview.. The word trigonometry is derived from the Greek words trigon and metron which means measuring the sides of a triangle. An angle is the amount of rotation of a revolving line with respect to a fixed line. If the rotation is in clockwise direction the angle is negative and it is positive if the rotation is in the anti-clockwise direction. Usually we follow two types of conventions for measuring angles, i.e., (i) Sexagesimal system (ii) Circular system. In sexagesimal system, the unit of measurement is degree. If the rotation from the initial to terminal side is th of a revolution, the angle is said to have a measure of 60. The classifications in this system are as follows: In circular system of measurement, the unit of measurement is radian. One radian is the angle subtended, at the centre of a circle, by an arc equal in length to the radius of the circle. The length s of an arc PQ of a circle of radius r is given by s rθ, where θ is the angle subtended by the arc PQ at the centre of the circle measured in terms of radians... Relation between degree and radian The circumference of a circle always bears a constant ratio to its diameter. This constant ratio is a number denoted by which is taken approximately as for all practical 7 purpose. The relationship between degree and radian measurements is as follows: right angle 80 radians radian (approx) radian radians (approx) 8/0/8

2 TRIGONOMETRIC FUNCTIONS.. Trigonometric functions Trigonometric ratios are defined for acute angles as the ratio of the sides of a right angled triangle. The extension of trigonometric ratios to any angle in terms of radian measure (real numbers) are called trigonometric functions. The signs of trigonometric functions in different quadrants have been given in the following table: I II III IV sin x + + cos x + + tan x + + cosec x + + sec x + + cot x Domain and range of trigonometric functions Functions Domain Range sine R [, ] cosine R [, ] tan R {(n + ) : n Z} R cot R {n : n Z} R sec R {(n + ) : n Z} R (, ) cosec R {n : n Z} R (, ).. Sine, cosine and tangent of some angles less than sine /0/8

3 6 EXEMPLAR PROBLEMS MATHEMATICS cosine tan 0 0 not defined n..6 Allied or related angles The angles ± θ are called allied or related angles and θ ± n 60 are called coterminal angles. For general reduction, we have the n following rules. The value of any trigonometric function for ( ± θ ) is numerically equal to (a) (b) the value of the same function if n is an even integer with algebaric sign of the function as per the quadrant in which angles lie. corresponding cofunction of θ if n is an odd integer with algebraic sign of the function for the quadrant in which it lies. Here sine and cosine; tan and cot; sec and cosec are cofunctions of each other...7 Functions of negative angles Let θ be any angle. Then sin ( θ) sin θ, cos ( θ) cos θ tan ( θ) tan θ, cot ( θ) cot θ sec ( θ) sec θ, cosec ( θ) cosec θ..8 Some formulae regarding compound angles An angle made up of the sum or differences of two or more angles is called a compound angle. The basic results in this direction are called trigonometric identies as given below: (i) (ii) (iii) (iv) sin (A + B) sin A cos B + cos A sin B sin (A B) sin A cos B cos A sin B cos (A + B) cos A cos B sin A sin B cos (A B) cos A cos B + sin A sin B tan A + tan B (v) tan (A + B) tan A tan B (vi) tan (A B) tan A tan B + tan A tan B 8/0/8

4 TRIGONOMETRIC FUNCTIONS 7 cot A cot B (vii) cot (A + B) cot A + cot B cot A cot B + (viii) cot (A B) cot B cot A tan A (ix) sin A sin A cos A + tan A (x) cos A cos A sin A sin A cos A tan A (xi) tan A tan A (xii) (xiii) sin A sin A sin A cos A cos A cos A (xiv) tan A tana tan A tan A tan A + tan A (xv) cos A + cos B (xvi) cos A cos B (xvii) sin A + sin B A + B A B cos cos A + B B A sin sin A + B A B sin cos (xviii) A B A B sin A sin B cos sin (xix) sin A cos B sin (A + B) + sin (A B) (xx) cos A sin B sin (A + B) sin (A B) (xxi) cos A cos B cos (A + B) + cos (A B) (xxii) sin A sin B cos (A B) cos (A + B) (xxiii) A + if lies in quadrants I or II A cos A sin ± A if lies in III or IV quadrants 8/0/8

5 8 EXEMPLAR PROBLEMS MATHEMATICS (xxiv) (xxv) A + if lies in I or IV quadrants A + cos A cos ± A if lies in II or III quadrants A + if lies in I or III quadrants A cos A tan ± + cos A A if lies in II or IV quadrants Trigonometric functions of an angle of 8 Let θ 8. Then θ 90 θ Therefore, sin θ sin (90 θ) cos θ or sin θ cos θ cos θ Since, cos θ 0, we get sin θ cos θ sin θ or sin θ + sin θ 0. Hence, ± + 6 ± sin θ 8 Since, θ 8, sin θ > 0, therefore, sin Also, cos8 sin 8 6 Now, we can easily find cos 6 and sin 6 as follows: Hence, cos 6 6 cos 6 sin Also, sin 6 cos Trigonometric equations 0 Equations involving trigonometric functions of a variables are called trigonometric equations. Equations are called identities, if they are satisfied by all values of the 8/0/8

6 TRIGONOMETRIC FUNCTIONS 9 unknown angles for which the functions are defined. The solutions of a trigonometric equations for which 0 θ < are called principal solutions. The expression involving integer n which gives all solutions of a trigonometric equation is called the general solution. General Solution of Trigonometric Equations (i) (ii) (iii) If sin θ sin α for some angle α, then θ n + ( ) n α for n Z, gives general solution of the given equation If cos θ cos α for some angle α, then θ n ± α, n Z, gives general solution of the given equation If tan θ tan α or cot θ cot α, then θ n + α, n Z, gives general solution for both equations (iv) The general value of θ satisfying any of the equations sin θ sin α, cos θ cos α and (v) (vi) tan θ tan α is given by θ n ± α The general value of θ satisfying equations sin θ sin α and cos θ cos α simultaneously is given by θ n + α, n Z. To find the solution of an equation of the form a cosθ + b sinθ c, we put a r cosα and b r sinα, so that r a + b and tan α b a. Thus we find a cosθ + b sinθ c changed into the form r (cos θ cos α + sin θ sin α) c or r cos (θ α) c and hence cos (θ α) c. This gives the solution of the given r equation. Maximum and Minimum values of the expression Acos θ + B sin θ are A + B and A + B respectively, where A and B are constants.. Solved Examples Short Answer Type Example A circular wire of radius cm is cut and bent so as to lie along the circumference of a hoop whose radius is 8 cm. Find the angle in degrees which is subtended at the centre of hoop. 8/0/8

7 0 EXEMPLAR PROBLEMS MATHEMATICS Solution Given that circular wire is of radius cm, so when it is cut then its length 6 cm. Again, it is being placed along a circular hoop of radius 8 cm. Here, s 6 cm is the length of arc and r 8 cm is the radius of the circle. Therefore, the angle θ, in radian, subtended by the arc at the centre of the circle is given by θ Arc 6.. Radius 8 8 Example If A cos θ + sin θ for all values of θ, then prove that A. Solution We have A cos θ + sin θ cos θ + sin θ sin θ cos θ + sin θ Therefore, A Also, A cos θ + sin θ ( sin θ) + sin θ sin θ + sin θ + Hence, A. Example Find the value of cosec 0 sec 0 Solution We have cosec 0 sec 0 sin 0 cos 0 cos 0 sin 0 sin 0 cos 0 cos 0 sin 0 sin 0 cos 0 sin 60 cos 0 cos60 sin 0 sin 0 sin (60 0 ) sin 0 8/0/8

8 TRIGONOMETRIC FUNCTIONS Example If θ lies in the second quadrant, then show that Solution We have sin θ + sin θ + sec θ + sin θ sin θ sin θ + sin θ + + sin θ sin θ sin θ + sin θ + sin θ sin θ cos θ (Since α cos θ α for every real number α) Given that θ lies in the second quadrant so cos θ cos θ (since cos θ < 0). Hence, the required value of the expression is secθ cos θ Example Find the value of tan 9 tan 7 tan 6 + tan 8 Solution We have tan 9 tan 7 tan 6 + tan 8 tan 9 + tan 8 tan 7 tan 6 tan 9 + tan (90 9 ) tan 7 tan (90 7 ) tan 9 + cot 9 (tan 7 + cot 7 ) () Also tan 9 + cot 9 Similarly, tan 7 + cot 7 Using () and () in (), we get tan 9 tan 7 tan 6 + tan 8 sin 9 cos 9 sin8 sin 7 cos 7 sin cos6 sin 8 cos6 + () () Example 6 Prove that sec8 θ tan8 θ sec θ tan θ Solution We have sec8 θ sec θ ( cos8 θ) cos θ cos8 θ ( cos θ) sin θ cos θ cos8 θ sin θ 8/0/8

9 EXEMPLAR PROBLEMS MATHEMATICS sin θ ( sin θ cos θ) cos8 θ sin θ sin θ sin8 θ cos8 θ sin θ sin θ cos θ sin8 θ cos8 θ sin θ tan8 θ tan θ Example 7 Solve the equation sin θ + sin θ + sin θ 0 Solution We have sin θ + sin θ + sin θ 0 or (sin θ + sin θ) + sin θ 0 or sin θ cos θ + sin θ 0 or sin θ ( cos θ + ) 0 or sin θ 0 or cos θ When sin θ 0, then θ n or θ n When cos θ cos, then θ n ± which gives θ (n + ) or θ (n ) or θ n ± All these values of θ are contained in θ is given by {θ : θ n, n Z} Example 8 Solve tan x + sec x for 0 x Solution Here, tan x + sec x which gives tan x ± n, n Z. Hence, the required solution set 8/0/8

10 TRIGONOMETRIC FUNCTIONS If we take tan x, then x 7 or 6 6 Again, if we take tan x, then x or 6 6 Therefore, the possible solutions of above equations are 7 x,, and where 0 x Long Answer Type Example 9 Find the value of 7 + cos + cos + cos + cos Solution Write 7 + cos + cos + cos + cos cos + cos + cos + cos cos cos 8 8 sin sin 8 8 cos cos cos + cos cos 8 Example 0 If x cos θ y cos (θ + ) z cos ( θ + ), then find the value of xy + yz + zx. 8/0/8

11 EXEMPLAR PROBLEMS MATHEMATICS Solution Note that xy + yz + zx xyz + + x y z. If we put x cos θ y cos (θ + ) z cos θ + k (say). Then k x cos θ, y k cos θ + and z k cos θ + so that + + cos cos cos θ + θ + + θ + x y z k [cos cos cos sin sin θ + θ θ k + cos θ cos sin θ sin ] [cos θ + cos θ ( ) k sin θ cos θ + sin θ ] 0 0 k Hence, xy + yz + zx 0 Example If α and β are the solutions of the equation a tan θ + b sec θ c, ac then show that tan (α + β) a c Solution Given that atanθ + bsecθ c or asinθ + b c cos θ Using the identities,. sin θ θ θ tan tan and cos θ θ θ + tan + tan 8/0/8

12 TRIGONOMETRIC FUNCTIONS We have, θ θ a tan c tan + b θ θ + tan + tan or (b + c) tan θ + a tan θ + b c 0 Above equation is quadratic in tan θ and hence tan α and tan β are the roots of this equation. Therefore, tan α + tan β a b + c α β b c and tan tan b + c α β α β tan + tan Using the identity tan + α β tan tan α β We have, tan + Again, using another identity a b + c b c b + c a a... () c c tan α + β α + β tan, α + β tan a c tan α + β a c We have ( ) Alternatively, given that a tanθ + b secθ c ac a c [From ()] 8/0/8

13 6 EXEMPLAR PROBLEMS MATHEMATICS (a tanθ c) b ( + tan θ) a tan θ ac tanθ + c b + b tan θ (a b ) tan θ ac tanθ + c b 0... () Since α and β are the roots of the equation (), so ac tanα + tanβ a b and tanα tanβ c a b b Therefore, tan (α + β) tan α + tanβ tan α tanβ ac a b c a b b ac a c Example Show that sin β + cos (α + β) sin α sin β + cos (α + β) cos α Solution LHS sin β + cos (α + β) sin α sin β + cos (α + β) sin β + (cos α cos β sin α sin β) sin α sin β + (cos α cos β sin α sin β) sin β + sin α cos α sin β cos β sin α sin β + cos α cos β sin α sin β sin β + sin α sin β sin α sin β + cos α cos β sin α sin β ( cos β) ( sin α) ( sin β) + cos α cos β ( cos β) ( cos α) ( cos β) + cos α cos β cos α Example If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and φ is their difference, then show that sin θ k + k sin φ Solution Let θ α + β. Then tan α k tan β 8/0/8

14 TRIGONOMETRIC FUNCTIONS 7 tan α or tanβ k Applying componendo and dividendo, we have tan α + tanβ tan α tanβ k + k sin α cosβ + cosα sinβ or sin α cosβ cosα sinβ k + k i.e., sin ( α + β) sin ( α β ) k + k Given that α β φ and α + β θ. Therefore, sin θ sin φ k + k or sin θ k + k sin φ Example Solve cos θ + sin θ Solution Divide the given equation by to get cos θ + sin θ or cos cosθ + sin sin θ cos 6 6 or cos cos or cos θ θ cos 6 6 Thus, the solution are given by, i.e., θ m ± + 6 Hence, the solution are θ m + + and m +, i.e., θ m and θ m Objective Type Questions Choose the correct answer from the given four options against each of the Examples to 9 Example If tan θ, then sinθ is 8/0/8

15 8 EXEMPLAR PROBLEMS MATHEMATICS (C) but not (B) or but not (D) None of these Solution Correct choice is B. Since tan θ is negative, θ lies either in second quadrant or in fourth quadrant. Thus sin θ if θ lies in the second quadrant or sin θ, if θ lies in the fourth quadrant. Example 6 If sin θ and cos θ are the roots of the equation ax bx + c 0, then a, b and c satisfy the relation. a + b + ac 0 (B) a b + ac 0 (C) a + c + ab 0 (D) a b ac 0 Solution The correct choice is (B). Given that sin θ and cos θ are the roots of the equation ax bx + c 0, so sin θ + cos θ b a and sin θ cos θ c a Using the identity (sinθ + cos θ) sin θ + cos θ + sin θ cos θ, we have b c a + a or a b + ac 0 Example 7 The greatest value of sin x cos x is (B) (C) (D) Solution (D) is the correct choice, since sinx cosx sin x, since sin x. Eaxmple 8 The value of sin 0 sin 0 sin 60 sin 80 is (B) (C) (D) 6 8/0/8

16 TRIGONOMETRIC FUNCTIONS 9 Solution Correct choice is (C). Indeed sin 0 sin 0 sin 60 sin 80. sin 0 sin (60 0 ) sin ( ) (since sin 60 ) sin 0 [sin 60 sin 0 ] sin 0 [ sin 0 ] [sin 0 sin 0 ] (sin 60 ) 6 8 Example 9 The value of cos cos cos cos is (B) 0 (C) 6 Solution (D) is the correct answer. We have 8 cos cos cos cos 8 sin cos cos cos cos sin 8 (D) 6 8 sin cos cos cos sin 8 sin cos cos sin 8/0/8

17 0 EXEMPLAR PROBLEMS MATHEMATICS 8 8 sin cos 8sin 6 sin 6sin sin + 6 sin sin 6sin 6 Fill in the blank : Example 0 If tan (θ ) tan (θ + ), 0 < θ < 90, then θ Solution Given that tan (θ ) tan (θ + ) which can be rewritten as tan( θ + ). tan( θ ) Applying componendo and Dividendo; we get tan ( θ + ) + tan ( θ ) tan ( θ + ) tan ( θ ) sin ( θ + ) cos ( θ ) + sin ( θ ) cos ( θ + ) sin ( θ + ) cos ( θ ) sin ( θ ) cos ( θ + ) sin θ sin 0 i.e., sin θ giving θ State whether the following statement is True or False. Justify your answer Example The inequality sinθ + cosθ holds for all real values of θ 8/0/8

18 TRIGONOMETRIC FUNCTIONS Solution True. Since sinθ and cosθ are positive real numbers, so A.M. (Arithmetic Mean) of these two numbers is greater or equal to their G.M. (Geometric Mean) and hence sinθ + cosθ sinθ cosθ sin θ + cos θ sin θ + cos θ sin + θ Since, sin +θ, we have sinθ+ cos θ sin θ + cos θ sin θ cos θ + Match each item given under the column C to its correct answer given under column C Example C C Solution (a) (b) (c) (d) cos x sin x + cos x cos x + cos x sin x (i) (ii) cot cot x x (iii) cos x + sin x x + sin x (iv) tan (a) cos x sin x x sin x tan x x. sin cos Hence (a) matches with (iv) denoted by (a) (iv) 8/0/8

19 EXEMPLAR PROBLEMS MATHEMATICS (b) + cos x cos x x sin cot x sin x. Hence (b) matches with (i) i.e., (b) (i) (c) (d) + cos x sin x x cos x cot x x. sin cos Hence (c) matches with (ii) i.e., (c) (ii) + sin x sin x + cos x + sin x cos x (sin x + cos x) ( sin x cos x) +. Hence (d) matches with (iii), i.e., (d) (iii). EXERCISE Short Answer Type tan A + sec A + sin A. Prove that tan A sec A + cos A. If sin α y, then prove that + cos α + sin α cos α + sin α + sin α is also equal to y. cos α + sin α cos α + sin α + cos α + sin α Hint:Express. + sin α + sin α + cos α + sin α. If m sin θ n sin (θ + α), then prove that tan (θ + α) cot α m + n m n [Hint: Express sin ( θ + α ) m sin θ n and apply componendo and dividendo]. If cos (α + β) and sin (α β), where α lie between 0 and, find the value of tanα [Hint: Express tan α as tan (α + β + α β] 8/0/8

20 TRIGONOMETRIC FUNCTIONS. If tan x b a + b a b a, then find the value of + a b a + b 6. Prove that cosθ cos θ cosθ 9 cos θ sin 7θ sin 8θ. [Hint: Express L.H.S. [cosθ cos θ 9θ cosθ cos ] 7. If a cos θ + b sin θ m and a sin θ b cos θ n, then show that a + b m + n 8. Find the value of tan 0. [Hint: Let θ, use θ θ θ sin sin cos θ sin θ tan θ θ + cos θ ] cos cos 9. Prove that sin A sina cos A cosa sin A. 0. If tanθ + sinθ m and tanθ sinθ n, then prove that m n sinθ tanθ [Hint: m + n tanθ, m n sinθ, then use m n (m + n) (m n)] p + q. If tan (A + B) p, tan (A B) q, then show that tan A pq [Hint: Use A (A + B) + (A B)]. If cosα + cosβ 0 sinα + sinβ, then prove that cos α + cos β cos (α + β). [Hint: (cosα + cosβ) (sinα + sinβ) 0]. If sin ( x + y ) a + b sin ( x y) a b Dividendo].. If tanθ, then show that tan tan x a [Hint: Use Componendo and y b sin α cos α sin α + cos α, then show that sinα + cosα cosθ. [Hint: Express tanθ tan (α ) θ α ]. If sinθ + cosθ, then find the general value of θ. 6. Find the most general value of θ satisfying the equation tanθ and cosθ. 8/0/8

21 EXEMPLAR PROBLEMS MATHEMATICS 7. If cotθ + tanθ cosecθ, then find the general value of θ. 8. If sin θ cosθ, where 0 θ, then find the value of θ. 9. If secx cosx + 0, where 0 < x, then find the value of x. Long Answer Type 0. If sin (θ + α) a and sin (θ + β) b, then prove that cos (α β) ab cos (α β) a b [Hint: Express cos (α β) cos ((θ + α) (θ + β))] m. If cos (θ + φ) m cos (θ φ), then prove that tan θ cot φ. + m [Hint: Express cos ( θ + φ ) m and apply Componendo and Dividendo] cos ( θ φ). Find the value of the expression [sin ( α ) + sin ( + α)] {sin 6 ( + α) + sin 6 ( α)]. If a cos θ + b sin θ c has α and β as its roots, then prove that tanα + tan β b a + c. tan θ [Hint: Use the identities cos θ + tan θ and sin θ tan θ + tan θ ].. If x sec φ tan φ and y cosec φ + cot φ then show that xy + x y + 0 [Hint: Find xy + and then show that x y (xy + )]. If θ lies in the first quadrant and cosθ 8, then find the value of 7 cos (0 + θ) + cos ( θ) + cos (0 θ) Find the value of the expression cos + cos + cos + cos [Hint: Simplify the expression to ( cos + cos ) 8 8 cos + cos cos cos /0/8

22 TRIGONOMETRIC FUNCTIONS 7. Find the general solution of the equation cos θ + 7sin θ Find the general solution of the equation sinx sinx + sinx cosx cosx + cosx 9. Find the general solution of the equation ( ) cosθ + ( + ) sinθ [Hint: Put r sinα, + r cosα which gives tanα tan ( 6 ) α ] Objective Type Questions Choose the correct answer from the given four options in the Exercises 0 to 9 (M.C.Q.). 0. If sin θ + cosec θ, then sin θ + cosec θ is equal to (B) (C) (D) None of these. If f(x) cos x + sec x, then f(x) < (B) f(x) (C) < f(x) < (D) f(x) [Hint: A.M G.M.]. If tan θ and tan φ, then the value of θ + φ is (B) (C) 0 (D) 6. Which of the following is not correct? sin θ (B) cos θ (C) sec θ (D) tan θ 0. The value of tan tan tan... tan 89 is 0 (B) (C) (D) Not defined 8/0/8

23 6 EXEMPLAR PROBLEMS MATHEMATICS. The value of tan + is tan (B) (C) 6. The value of cos cos cos... cos 79 is (D) (B) 0 (C) (D) 7. If tan θ and θ lies in third quadrant, then the value of sin θ is 0 (B) 0 8. The value of tan 7 cot 7 is equal to (B) + (C) (D) 9. Which of the following is correct? sin > sin (B) sin < sin (C) (C) sin sin (D) sin 8 sin 0 (D) 0 [Hint: radian If tan α approx] m m +, tan β, then α + β is equal to m + (B) (C). The minimum value of cosx + sinx + 8 is (B) 9 (C) 7 (D). The value of tan A tan A tan A is equal to (B) (C) (D) tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A None of these 6 (D) 8/0/8

24 TRIGONOMETRIC FUNCTIONS 7. The value of sin ( + θ) cos ( θ) is cosθ (B) sinθ (C) (D) 0. The value of cot + θ cot θ is (B) 0 (C) (D) Not defined. cos θ cos φ + sin (θ φ) sin (θ + φ) is equal to sin (θ + φ) (B) cos (θ + φ) (C) sin (θ φ) (D) cos (θ φ) [Hint: Use sin A sin B sin (A + B) sin (A B)] 6. The value of cos + cos 8 + cos 6 + cos is (B) (C) (D) 8 7. If tan A, tan B, then tan (A + B) is equal to (B) (C) (D) 8. The value of sin sin is 0 0 (B) (C) (D) + [Hint: Use sin 8 and cos 6 ] 9. The value of sin 0 sin 70 + sin 0 is equal to (B) 0 (C) 0. If sin θ + cos θ, then the value of sin θ is equal to (D) (B) (C) 0 (D) 8/0/8

25 8 EXEMPLAR PROBLEMS MATHEMATICS. If α + β, then the value of ( + tan α) ( + tan β) is (B) (C) (D) Not defined θ. If sin θ and θ lies in third quadrant then the value of cos is (B) 0 (C) (D). Number of solutions of the equation tan x + sec x cosx lying in the interval [0, ] is 0 (B) (C) (D). The value of sin + sin + sin + sin is given by sin + sin (B) 8 9 (C) cos + cos (D) cos + sin If A lies in the second quadrant and tan A + 0, then the value of cota cos A + sin A is equal to (B) The value of cos 8 sin is (C) 7 0 (D) (B) (C) (D) [Hint: Use cos A sin B cos (A + B) cos (A B)] 8/0/8

26 TRIGONOMETRIC FUNCTIONS 9 7. If tan α 7, tan β, then cos α is equal to sin β (B) sin β (C) sin β (D) cos β 8. If tan θ a, then b cos θ + a sin θ is equal to b a (B) b (C) 9. If for real values of x, cos θ x +, then x θ is an acute angle (B) θ is right angle (C) θ is an obtuse angle (D) No value of θ is possible Fill in the blanks in Exercises 60 to 67 : a b (D) None 60. The value of sin 0 sin 0 is. 6. If k 7 sin sin sin, then the numerical value of k is If tan A cos B sin B 6. If sin x + cos x a, then (i) (ii) sin 6 x + cos 6 x sin x cos x., then tan A. 6. In a triangle ABC with C 90 the equation whose roots are tan A and tan B is. [Hint: A + B 90 tan A tan B and tan A + tan B sin A ] 6. (sin x cos x) + 6 (sin x + cos x) + (sin 6 x + cos 6 x). 66. Given x > 0, the values of f(x) cos + x + x lie in the interval. 8/0/8

27 60 EXEMPLAR PROBLEMS MATHEMATICS 67. The maximum distance of a point on the graph of the function y sin x + cos x from x-axis is. In each of the Exercises 68 to 7, state whether the statements is True or False? Also give justification. 68. If tan A cos B sin B, then tan A tan B 69. The equality sin A + sin A + sin A holds for some real value of A. 70. sin 0 is greater than cos cos cos cos cos 6 7. One value of θ which satisfies the equation sin θ sin θ lies between 0 and. 7. If cosec x + cot x then x n, n + n 7. If tan θ + tan θ + tan θ tan θ, then θ If tan ( cosθ) cot ( sinθ), then cos θ ± 76. In the following match each item given under the column C to its correct answer given under the column C : (a) sin (x + y) sin (x y) (i) cos x sin y (b) cos (x + y) cos (x y) (ii) (c) cot + θ (d) tan + θ (iii) (iv) tan θ + tan θ + tan θ tan θ sin x sin y 8/0/8