The matrix: "Is an organization of some elements written in rows

Transcription

1 st term st Sec. Algebra. The matrix: "Is an organization of some elements written in rows Ex: and columns between brackets in the form ( ) ". st column nd rd - st row - nd row 7 rd row * The order of any matrix = no. of rows x no. of columns How to express the elements in the matrix. a a a A= a a row column Ex: a a a is the element in rd row and the nd column. Write the order of A, B and C then find: a, a, a, b, b, b, c and c 9 8 A=, B= / 8 and C = /

2 The order of A = x Answer a =, a = 9 a = The order of B = b = b = b = The order of C = C = C = C = Some types of matrices: -The raw matrix Ex: A = ( -) x matrix B = ( -9) x matrix -Column matrix Ex: X = - x matrix - Squared matrix Which no. Of rows = no. of column. - Ex: Y = -, - 8 -Zero matrix x matrix 8 All it is elements are zero x matrix Ex: A=, B = x x /

3 Ex: Write the matrix (A xy ) of the dimensions where a xy = x y row X =,, column Y =, Answer Note each element of X with all elements of Y a = = a = = a = = a = = a = = a = = A = Ex: Write the matrix (B xy ) of the order where b xy =x y EX: Write the matrix (B xy ) if the order x where bxy = x+y

4 EX: Write the matrix B = (b yz ) with the order x where Y + Z if Y>Z b yz if Y = Z Z Y if Y < Z Answer Y = Z = b = b = = b = = b = + = b = b = = b = + = b = + = b = B = Ex: Write the matrix A = a xy with the order x where a xy = Zero if X = Y X + Y a < b if X Y Ex: Write the matrix X = X ab with the order where Zero a = b X ab = a > b a + b a < b Transpose of matrix: If A = (a xy ) then A T = a yx Where A T is the transpose of A Note: (A T ) T = A

5 Ex: Find the transpose of the following matrices and write its order: 9-7 A -, B = - and C = D and L = ( ) / - - / Answer - A d = C d = X B d = (9 ) D d = x L d = - x Verify that (A d ) d = A The equality of two matrices If A and B are two matrices then A = B if and only if - A and B with the same order - The corresponding elements are equal. - - Ex: = - -

6 But: 8 and as as. Ex: Find the values of x, y and Z if 7 - X + = 7 y- - The two matrices are equal Z = - () y = 7 () X + = () y = 7 y = 7 + = y = y = = y = X + = X = = X = Ex: If X = Answer a + b h c + d 8 9, Y = 8 8 d + c

7 Find a, b, c, d and h if X = Y T 8 L X - Ex: If = Z - y y 8 Find the value of X, y, z and L. Ex: Find the value of X, y and Z which make the two matrices X+ y y Z and X + y Ex: Find the value of X, y which make A = B d where A = ( - ) B= y x equal Symmetric and skew symmetric matrices: If A is a square matrix, then A is called a symmetric matrix if and only if A = A T A is called a skew symmetric matrix if and only if A = - A T A= is symmetric matrix B= is skew symmetric 7

8 Sheet () I- Complete: - If A is a matrix of order x and if a =, a =, a = / and a =, then the matrix A =. - If X = -, then the matrix X is of order., X =, X =, X =.. - If O is Zero matrix of order x O T = - ) X = -, X = Y T then y =., y =. ) X = - = X -, then X =. - X= - -, Y = -7 8 / and Z = / - / - Mention the order of each matrix. - Write each of the following elements X, y, Z, X, y, Z - ]-If X = - Check that (X T ) T = X a X - ]- If = Z+y y Then find the value of each X, y, Z. 8

9 ]- From the matrix X = X ij of order where i + j if i > j X ij = o if i = j j i if i < j ]-From a matrix y = (y ij ) of order Y ij = i j + Then find the matrix X where X = Y T mention its order and find X ij if i=j. 7] show which of the following is symmetric and which is skew symmetric : a) b) 8] If A = x 8 x y is a symmetric matrix, then Find the value of : x, y 9] If B = x 7 x z y x is skew symmetric matrix Find the value of x, y and z 9

10 Operation on matrices I-Addition: To add two matrices A, B they must have the same order. Ex: If A =, B = -7 A + B = - (-7) = - 8 Ex: If A = - A = - = 8 - Properties of addition of matrices: Let A, B, C matrices of order m x n, o mxn be a zero matrix of the sam order. -Closure property: A+B is a matrix of order mxn -Commutative property: A+B = B+A Ex: Associative property: (A+B) + C = A + (B+C) Ex: If A -, B = - -, C = Then (A + B) + C =

11 = A + (B + C) = Identity of addition: A + O = O + A = A Ex: A + O = A -Additive inverse: A + (-A) = (-A) + A = Where A is the additive inverse of A A + ( A) = Ex: If A -, B =, C = Find: () A + B () B C () A + B C Answer () A + B = - (-) - - () B C = B + ( C)=

12 () A + B C = - = 8 - = ) ( ) ( 8 (-) = 7 = A + B + (-C) Ex: If X = Z, Y, Find () X Y + Z () X y + Z () X Y Z Ex: If A = - B, - - Find the matrix X such that A X = B

13 Answer A X = B X = B A X = B + A = A + ( ) B Find X by your self. Ex: If - d b c a - Find a, b, c, d Answer d b c a a = a = = a = b = b = = b = d = d = + = d = Find C. Ex: If A B, - 7 Find the matrix X if X = A B Ex: If B D, C, Find the matrix X which satisfies the equation B + C = X D.

14 Ex7: If A, B Find the matrix X such that: B + X T = A B + X T = A X T = A B X T = = = 7 (X T ) T = X Ex8: If X T + d b c a Find the matrix X Ex9: If A B, Find the matrix X if A + X = B

15 Sheet () I-Complete: ) I A + - =, then A = ) If O is the Zero matrix of order x, then O =.and it is of order. ) If each of the matrices A and B is of order x, then the resultant matrix of A B is of order ) T = which is of order ) If A =, then A =, - A =. ) If A =, then A = II- If A = - and B = - Check that: ) (A + B) T = A T + B T ) A B B A III- If A = - and (A + B) T = 7 - Find the matrix B. IV- If Y 7 X - 7 Find the value of X and Y.

16 V- Find X, Y, Z, and L that satisfy that: X L Z Y = O x VI- If A = and B = 7 Find the matrix X such that: ) A + X = B ) A + B X = ) A X = B T VII- If X + X T = 9, find the matrix X. VIII- If A = and B T =, find the matrix X such that: X B + A T = A + (B X) T.

17 7 Multiplying Matrices If A is a matrix of order mxn, B is a matrix of order r x L, then their product C = A x B will be defined if and only if n = r To multiply two matrices A no. of columns= no. or rows B x x Ex: If A= B, AB =, 9 ) x (- ) ( x ) x (- ) ( x (-x ) -) (x (- x ) ( x -) Ex: If A, B = 7, C = and D = Check that: ) (AB) T = B T A T ) (AB)C = A(BC) ) (B + D) C = BC + DC. ) D - D + I = (D I) Answer ) (AB) = 7 = 8 9 = x is the order of the product matrix

18 8 (AB) T = B T A T = 7 = 8 9 = ) (AB) C = = = A (BC) = 7 = 7 9 = 8 7 =

19 9 Sheet () Review on unit () I-Complete: -If 7 X - Y =, then X =.., Y = - If I is the identity matrix of order x then (I )t = - If T y X Z Then X =, Y =., Z=.. -If A is a matrix of order x, B is a matrix of order then AB is a matrix of order x. - = = If A = 9, B =, then A + B =.. 7-If I is the identity matrix of order x, then I = 8- If A = 8, then A T =, A T is of order =.. 9-If the matrix A is of order m x n, B is a matrix of order L x K, then the required condition that makes AB defined is.in this case the matrix AB is of order.. - =.. - If y + Z = X, then X =., y=.., Z=

20 - ( ) =. - If X = then X =. - + =. - If X + Y Z = 8 Then X =.., y =.., Z =.. - If A =, then (A T ) T = 7- + =. 8- If X + = II- a) If C =, Q T =, A=( - )and B = Prove that AB = QC T. b) IA =, B =, C =

21 Q = Prove that C (A + B T ) Q = I c) Solve the matrix equation: T X = X T I d) If A =, B = 8 7 Prove that: (AB BA) + I = e) If A = -, B = 8 and C = Prove that: AB C T = I F) If: A=, B = ( ), Find the matrix X such that X T = (AB) g) If A = X and (A T ) A T I = h) If A and B are two matrices and if BA= Find A T B T. i) If L =, M = and Q T = find the matrix: L (M T - Q) k) A =, B =, n is a realnumber Prove that: (A + nb ) = ni where I is the identity matrix of order x.

22 L) If A =, B T = Find the value of X where (X ) I = A + B m) If A prove that A A + A I = (A- I) n) If (A T B) T =, B = find the matrix A. o) If A =, B T = Find the value of X where (X ) I = A + B. p) A T = 7, B = 7 Prove that (AB) = 7I. r) If A =, B =, C = and if [X T AB] = C T find the matrix. S) If X =, Y T = Prove that: XY = YX T) A = -, B = Prove that: A T + B T = I V) If A =, Prove that A + I =

23 second order Determinants a b A ad cb c d Find the value of the following determinant : a) 8 b) 7 c) d) sin cos cos sin Third order a b c e f d f d e A d e f a b c h j g j g h g h j e f d f d e a b c h j g j g h Find the value of the following determinant : a) b) solve the equation : x 8 x x x x Find the area of a triangle whose vertices are X(,),Y(,-) and Z(-,) Prove using determinants that the point (-,), (,) And (8,-) are collinear. Solve the system of the following equation using Cramer s Rule : x+y=-, x y = -

24 Multiplicative inverse of a matrix ] Show the matrix which have multiplicative inverse : a) b) c) d) e) f) ] what is the real values of a which make each of the following matrices has A multiplicative inverse : a) a b) a 9 a ] if : X = x x prove that : X - = X ] solve each of the following system using the matrices : a) x+y=, x+y= b) x-7y=, x-y= ] use the matrices to find the two numbers in which their sum equal and The difference between them equal.

25 Solving inequatities of first Degree in one variable and in two variables Show: graphically the solution set of the inequality X+>, then write in the form of an interval where X R Solution: X + > X> X>-9 X > - S.S = ] -, [ ) X X < X + X X X < X + X X - + X X< + -X (-) X < X - X < S.S = [-, [ S.S = ] -, [ S.S. of the original inequality = [-, [ n ]-, [ = [-, [

26 - Solve the inequality graphically: a) X Y Step (): X Y = Put X = Step (): Put Y = Step (): Graph. Step (): Substituted by (,) < X Y - b) X + Y < Step (): X + Y = Step (): Step (): Test (, ) < X Y

27 Note: -The equation: Y = is represented by the X axis. -The equation: X = is represented by the Y axis. -The equation: Y = a is represented by a straight line parallel to the x- axis and passes through the point (, a). -The equation X = a is represented by straight line parallel to y- axis and passes through the point (a, ). -The straight line whose equation is in the form: X Y passes a b through two points (a, o) and (o, b). Ex: Represent graphically the S.S of the two inequalities X -y, X + y Answer Let X y = Let X + y = Test (, ) Test (, ) < < X Y - / X Y - / S.S is colloured region S = S S 7

28 Find graphically the S.S of each of the following: X, y, X + y, X + Y Answer L: X = L: y = L : X + y = X Y Test + (, ) O < L: X + y = X Y Test (, ) < S.S is coloured region. 8

29 Linear programming Find graphically the S.S of the set of inequalities: - X, y, y- X and y + X Then find from the S.S the value of (X,y) that makes R maximum value where R = X + y Answer L: X =, L: y = L: y-x = L: y + X = Test + (, ) Test + (, ) O < O < X - Y X Y 9

30 Points of intersection are (, ), (, ), (, ), (, ) sub in R at (,) X + y = X + X = at (, ) Max value at (, ). + = + 9 = 9 At (, ) + = Mx. Ex: Find the greatest and smallest value of the expression L = X+y- on the region that satisfies the following condition: -X ; - y, X + y and X + y - L: X + y = X Y L: X + y = - X - Y - L = X + y at (, ) : L = + = - at (, -) L= = - at (, -) L = =-7 at (-, ) L = - = -8 at (-, ) L =-+ = at (, ) L = + = 7 Smallest value is 7 Greatest value is 7.

31 Ex: A factory produces two kinds of accessories (A) and (B) producing one piece of the first kind (A) needs two machines. The first machine works for one hour second machine works for hours and half and producing one piece from the nd kind (B) needs the first machine to work for hours and the second machine for hours if the first machine works for 8hours. At most daily and the second machine works hours at most daily. The profit of the factory is L.E. and L.E from each piece of the kind (A) and (B) respectively. Find the maximum profit that the factory can make in one day. Answer -Summarize the data in the problem in the following table: (A) (B) The most no. of hours First machine needs no. of hours 8 Second machine needs no. of hours. / The profit - Translate the data in the form of inequalities. ) X, y ) X + y 8 ) / X + y X X + y The objective function. R = X + y.

32 - Represent the inequalities graphically. L: X + y = 8 L: X + y = X 8 Y X.8 Y Test (, ) Test (, ) < 8 < A = (.8,), B = (,), C = (,), D (, ) The objective function R = X + y R A = X.8 + =. R B = X + X = R C = + X = 8 R D = Zero. The maximum profit = when the factory produces units from kind A and one unit from kind B.

33 I- Choose: -In the S.S of the inequalities: Review sheet () On linear programming X, y, X + y 8, X + y the point (.,.) is the point that makes the objective function R = X+y be maximum. a) (,) b) (,) c) (,8) d) (,) -The point lies in the region of the s.s of the inequality X + y 8. a) (,) b) (-,) c) (,) d) (,) -In the S.S of the inequalities X, y, X + y, X +y we find that (, ). Makes the objective function R=X+y. give the maximum value. a) (., ) b) (,) c) (,) d) (,) -In the S.S of the inequalities X, y,x+ y,x+ y we find (, ) makes the value of the objective function R = X + y minimum. a) (,) b) (,) c) (,) d) (,) - The point..belongs to S.S of the inequality X. a) (, ) b) (, ) c) (-, ) d) (-, ) -The point (, -) belongs to the S.S of the inequality: X+y... a) b) c) > d) = II- Complete: -The point (,) belongs to the S.S of the inequality X + y. -(,) belongs to the S.S of the inequalities X -, y

34 -(,) belongs to the S.S of the inequality: y- X. -(,) belongs to the S.S of the inequality: y.. X. -(,-) belongs to the S.S of the inequality: y + X Zero. III- Find graphically the S.S. of the two inequalities: ) y X, X < < simultaneously. ) y X, X y < simultaneously. ) Find graphically the S.S of y, y X ) y X, X y > ) X, y, y X, X + y ) y, y X IV- a) A carpentry work shop produces two kinds A and B. Each table is made by two persons, a carpenter and a painter. The carpenter needs one hour to make the table of the kind A while he needs two hours to make the table of kind B the painter needs two hours to paint the table of kind (A) and one hour to paint the table of kind (B). the carpenter works for 8 hours daily at most while the painter works for hours daily at most. If the work shop sells all it daily production with profit L.E to the table of kind A and L.E to the table of kind B. How many tables of each kind the work shop should produce daily to achieve maximum profit. b) A baby home decided to offer a light meal to babies the meal consists of kinds of pies such that the meal given to each child contains units at least of vitamin A and 9 units at least of vitamin B. If we assume that the pie of the first kind gives at average one unit of vitamin A and units of vitamin B and the price of the pie of the st kind is one pound and the price of the pie of the nd kind. pound, find how many pies of each kind needed to make the meal cheaper and guarantees. The Lowest limit of vitamins.

35 Chapter () Measure of angles Directed angle: "Is an ordered pair formed from two rays having the same starting point that is called the vertex of the angle". B Terminal side ( OA, OB) O Intial side A (OB, OA) Terminal side B O Intial side A (OA, OB) (OB, OA) The positive and negative measures of the directed angle. ) The measure of the directed angle is +ve if the direction of it is antilock wise rotation. ) The measure of the directed angle is ve if the direction of it is a clock wise rotation.

36 * Angles in the standard position: X \ y X X \ y X nd quad X \ B y X X \ y st quad B X y \ B th quad B rd Quad y \ y \ y \ 7 < θ < 8 < θ <7 9 < θ <8 < θ <9 Note: Each angle in its standard position has two measures. One of them is positive and the other one is negative. a) If X is positive measure of a directed angle then its negative measure = X. Ex: =. b) If ( - X ) is the negative measure of a directed angle, then its positive measure = X. Ex: =. Ex: Determine the negative measure of each of the following angles: ) 7 ).7π ) \ \\ * Determine the positive measure of each of the following angles: ) -.8 ) / 9 π ) \ \\ * Determine the quadrant in which the following whose measures as follows, lies ) Answer = lies in nd quad lies in nd quad 7

37 ) 7 Then get the positive measure 7 - = 7 lies in th quad - 7 lies in th quad. ) 7 ) * Systems for measuring angles. ) Degree measure system. ) Radian measure system. θ rad = The length of the arc The radius of the circle L θ rad = r L θ rad r Ex: Find the radian measure of angle that subtends an arc of length L in a circle its radius (r). ) L = π cm, r = cm Answer θ rad = L r π π 7 ) L =.7cm, r = 9.7cm. Ex: If r = cm, θ rd =. find L. L = r θ rad =. = 8. * The relation between the degree measure and the radio measure. X 8 θ rad π 7

38 EX: Find the degree measure and radian measure of the central angle that subtends an arc of length (L) in a circle of radius (r): ) L = π, r = cm. θ rad = L r X 8 π, π X 8 8 X = π ) L = cm, r = 7cm ) L = cm, r = cm. π π Ex: ABC is a triangle in which: m (<A) =, m (<B) = find m(<c) in radian. m (<C) = 8 ( + ) = 7 X 8 θ rad = 7 8 θ rad π θ rad π ( π ) 7 rad. Answer Ex: ABC is a triangle in which m (<A) = and m (<B) = Find the measure of third angle in degree and radians. Answer 8

39 Ex: Find the circumference of a circle in which an arc with length X 8 θ rad = r = cm, and opposite to an inscribed angle with measure. 8 L θ rad θ rad π rad = π = Error! Not a valid link. circumference of a circle = π r = xπ X π 8 Circumference = πr or πd = 9cm. Ex: Find the area of the circle that the central angle drawn in it whose measure is and subtends an arc with length πcm. r = L θ rad Area of a circle = πr Ex: The sum of measures of two angles 7 and the difference between them is π. Find the measure of each angle in degrees and radian. Let the measure of st angle = X Answer Let the measure of nd angle = y Their sum = 7 X + y = 7 () Their diff = π 8 X y = () Add () and () X + y = 7 + X y = Add X = 9

40 X = = Subin () X + y = 7 + y = 7 y = 7 = 7 X 8 X rad = Y rad = X rad π 8 7 π 8 rad.9 rad.9 Ex: Find in radian and degrees the measure of central angle that s subtends an arc of length 8cm in a circle of diameter cm long. Ex: Find the area of the circle that the central angle drawn in it whose measure is and subtends an arc with length π cm. Ex: The ratio among the measures of the interior angles of a triangle is :: find the measures of its angles in degrees and radian.

41 I-Complete: Sheet () Review on chapter () -The radian measure of the sum of measures of the interior angles of a quadrilateral = -The angle whose measure = in the standard position is equivalent to the angle whose measure is. -The least positive measure of the angle of measure (-9 ) equals.. -If the arc length in a circle equals its circumference, then the 8 measure of the central angle that subtends this arc equals -If we have an angle with radian measure θ rad and its degree measure is X then X π = -The angle whose measure is lies in.quad. 7-In ABC : m (<A) = and m (<B) = 7. then m (<C) = 8 rad. 8-The radian measure of a central angle in a circle of radius length cm and subtended an arc of length 8cm =.rad. 9-The arc whose length is π. In a circle of radius length 9cm. is opposite to a central angle of degree measure =. - 8 \ =.rad. II- A) find the radian measure and degree measure of a central angle subtended an arc of length cm. in a circle of radius cm. Long....

42 B) Find the length of the arc that is intercepted by a central angle of measure in a circle with diameter length = cm.... C) ABC is a triangle in which m (<A) = 7 and m (<B) = find m(<c) in radian and degrees.... D) The difference between the measures of two supplementary angles is find the measure of each of them in degrees and radians.... E) A circle with radius length 7.cm is circumcircle of ABC if m(<bac)= and m (<ABC)=, find the length of the three arcs that the circle is divided by the vertices of ABC....

43 I-The unit circle: Chapter () Trigonometric functions It's a circle its centre is O the origin point (, ) and its radius is unit length. * X [-,], y [-, ] X \ (, ) B (x, y) θ y X (-, ) O x A (, ) y (, -) * X + y = y \ II- The basic trigonometric functions and their reciprocals. Let m < A o B = θ The F n X = Cos θ Y = Sin θ then its reciprocals X y = = Sec θ Cos θ = Sin θ = Cosec θ y X = tan θ = = Cot θ X y Tan θ III- The sign of the trigonometric functions: Sin, Cosec (+ve) The all are (+ve) tan, Cot (+ve) Cos, sec (+ve) A S T C

44 Ex: Determine the signs and the quadrant of the following trigonometric functions. - Cos ( - ) Answer * Step () get the positive measure = - * Step (): = 9 9 lies in rd quad - lies in rd quad *Step (): Cos is ve in rd quad Cos ( ) is ve in rd quad. Ex: Cot Step () Step (): Step (): Ex: Sin Cot Cot Answer = Cot ( 8) = Cot 9. = Cot 9 9 > Cot 9 = cot ( + ) lies is rd quad, Cot is + ve in rd quad Cot is +ve, Ex tan 78, Ex Sec ( ) 9 7

45 Ex: If the termind side of the angle of measure θ in its standard 7 position cuts the unit circle at the point A (, ) ) Determine the quadrant in which the angle lies. ) Find all the trigonometric functions O f θ. Answer ) rd quad (Why) ) X =, Y = 7 X = Cos θ =, Sec θ = = Cos θ, Y = Sin θ = 7, Cosec θ = Y Sin θ = tan θ = = X Cos θ = = 7 Cot θ = = tan θ 7 # Ex: If the terminal sides of the angles of measure θ in its standard position cuts the unit circles at the point B (, ) : ) Determine the quadrant in which the angle lies. ) Fin all the trigonometric functions of θ. Ex: If θ, is the measure of a directed angle in its standard position, B is the point of the intersection of its terminal side with the unit circle, find all the trigonometric functions of θ: ) B ( X, -.), X > O Answer X + Y = X + (.) = X +. = X =. =. X =.

46 X = ±. = ±.8 But X > X =.8 X =.8, Y =. ( th quad) Cos θ =, Sec θ =, Sin θ =, Cosec θ =. Tan θ =.., Cot θ =. ) B ( - X, X ) X > ) ( a, -a) where 7 < θ < Ex: If θ ], [ and cos θ =, find the value of cosec θ sinθ tan θ cosec θ Answer: Cos θ = X = X + y = ( ) Y Y = - Y = Y = ± But θ ], [ in st quad Then X and y must be +ve Y =, X = Cosec θ sin θ = Y tan θ = = X = = tan θ Cosec θ = Cose θ sin θ tan θ Cosec θ = 9 = ± Cosec θ = Sin θ =

47 The trigonometric functions Of some special angles 9 (,) 8 (-,) (,) (,-) 7 The measure of angle The point The F n Cosθ Sinθ Tanθ Sec θ Cosec θ Cot θ ) O or : (,) Undif Undif ) 9 (,) Undefined Undif ) 8 (-,) - - Undif Undif ) 7 (,-) - Undefined Undif - ), ) (, ) 7), 7

48 Verify each of the following: ) sin Cos tan = ) Cos. tan sec cosec = ) tan tan tan tan = Find the value of X if: a) X sin cos π = tan sin b) tan X = Cos sin cos 8 sing 7. 8

49 Find the value of: ) Sin Step (): in nd quad Sin is +ve in nd quad. Sin is +ve. Step (): Step (): Sin = Sin (9 + ) or Sin (8 ) Sin (9 + ) = cos = Properties of trigonometric Function S. Answer Or Sin (8 - ) = Sin = Changes Note () Note () Cos (-θ) = Cos θ Sec (-θ) = Sec θ Sin Cos Sin (-θ) = - Sin θ Cosec (-θ) = -Cosec θ Tan Cot Tan (-θ) = - tan θ Cot (-θ) = - Cot θ Sec Cosec Ex: Find the value of: ) tan (- )= - tan st quad tan +ve change ) Sec ( ) = - tan (9 - ) = - cot + = Change 8 Not change 9

50 Answer: ) Sec ( ) = Sec ( ) = Sec Sec = - Sec (8 ) = - Sec = - ) Cot (-78 ) = - cot (78 ) = - cot (9 ) = - tan = - nd Sec -ve quad Not change 78 7 ) Sec () ) tan () ) Cosec 7) Cos (9) 8) Cosec 9) Sin ( 7 ) Find the value of: ) sin cos ( - ) + sin sin (- ) * sin = sin (8 ) = sin = *cos (- ) = cos = Answer * sin = - sin (7 + ) th quad = - cos = - *Sin (- ) = - sin = - sin (8 ) nd quad = - sin = -

51 + (- ) (- ) = + = ) Sin tan + cos (- ) cosec ( ) ) Cos sin sin cos (- ) ) sin 9 cos cos (-8 ) sin ( ) Ex: If the terminal side of a positive angle whose measure is θ in the standard position cuts the unit circle at the (, y) find ) tan (9 θ) + sec (9 θ) ) cot (7 + θ) tan (θ 8) ) cos ( ) sin (θ π) ) cos ( - θ) + θ) X + y = Answer ( ) + y = 9 + y = y = - 9 = y = ) tan (9 θ) + sec (9 θ) = cot θ + cosec θ = ( ) + = + = B (, )

52 ) cot (7 + θ) tan (θ 8 ) - tan θ tan (θ 8 + ) - tan θ tan (8 + θ) - tan θ tan θ = - ( ) ( ) = = 8 Ex: If cosc = where 8 < C < 7, find the value of each of: ) cosec (8 + C) ) cot (C 9 ) ) sec ( C) ) sec (9 + C) ) tan (7 C) ) cot (7 + C) X = X + y = ( Y = Y = ( ) + y =, 9 = ) ) Cosec (8 + C) = - Cosec C ) Cot (C 9 ) = ( ) = = cot (C 9 + ) = cot (7 + C) = tan C = ( ) Sec ( C) = Sec C = ) =

53 Ex: Find one of values of X that satisfies each of the following where < X 9. ) Cosec (X + ) = Sec (X ) Answer X + + X = 9 X + = 9 X = 9 - X = 7 X = 7 = ) cot (X ) = tan (9 + X) ) sin (X + ) = cos (X + ) ) tan (X + 8 \ ) = cot (X + \ ) Ex: Find the S.S of each of the following equations given X ],π [ ) sin X = Answer sin X = + sin X = sin X = X = or X = 8 = S.S = {, } ) tan X + = tan X = is +ve in st quad or nd quad Answer is ve in nd, th quads X = 8 = Or X = = S.S = {, }

54 ) sin X = sinx = is +ve in st or nd ve in rd or th = ± X = or X = 8 = Or X = 8 + = Or X = = S.S = {,,, } ) cos X + = ) sec X = ) tan (9 X) = 7) tan X + tan = Notes: Cos (-θ) = Cos θ Sec (-θ) = Sec θ Sin (-θ) = - Sin θ Cosec (-θ) = -Cosec θ Tan (-θ) = - tan θ Cot (-θ) = - Cot θ If sin C = cos d If cosec C = sec d If tan C = cot d C ± d = 9 + n, where n Z C ± d = 9 + n, where n Z C + d = n

55 Ex: Find the value of X: < X 9 ) sin (X + ) = cos (X + ) Answer ) tan (X + ) = cot (X ) ) sec (X + ) = cosec (X )

56 Opposite Values of trigonometric functions of an acute angle In a right angled triangle: sin θ = cos θ = tan θ = AB AC BC AC Opposite Adjacent cosec θ = AC AB sec θ = AC BC cot θ = BC AB Opposite Hypotenuse Adjacent Hypotenuse AB BC Hypotenus e C θ adjacent A B Ex: Complete: X ) cot ( 8 + Z) = ) tan ( 8 Z) = 7 ) cos ( 9 + Z) = ) sec ( 7 X) = Z y ) cosec ( 9 X) = Ex: If sin X = and sin Y = where X and Y are two measures of two positive acute angles, find the value of sin(8 X) cos tan (9 X) sec (9 θ) cos ( (8 θ) sec (8 θ) X)

57 Ex: XYZ is a right angled triangle at Y. If YZ = XY, find the value of each of: tan Z, tan X, cos Z, cosec X and sin Z. Ex: ABC is a right angled at A, AB = 7cm draw AD BC where D BC and BD =.9 cm. Calculate the value of each of: sinc, tan B, sec (<DAC) and cosec (<BAD). 7

58 On finding the measure of an angle given the value of one of its trigonometric function Ex: If cos θ = where 7 < θ <. Find the expression sin (8 θ) + tan (9 θ) tan (7 θ). Ex: If tan A = where A is the measure of a positive acute angle, find the value of the expression cos sin cosec sec cos (9A) cot cosec (8 (8 A) A) 8

59 A- Complete: - If X Final revision I- Algebra and trig. y = Then X =. and Y =. - The point (, ) belong to the S.S of the inequality X + Y. - The angle whose measure ( - ) lies in quadrant. - If sec X =,cot X < X ], [ then X =. - If cosec X = sec X where < X < 9, then sin X= - The degree of a central angle that subtends an arc of length cm. in a circle of radius length = cm is 7- If tan X = where X ] π, - [ then cos (7 + X) =. 8- If A is a matrix of order, B is a matrix of order.. then AB is a matrix of order. 9- The sum of measures of the interior angles of a triangle in radians =.. - If sec θ =, tan θ =. Or θ=. Where <θ<π. - If the points: A (, ), B (, ), C (, 9) belong to the S.S of the inequalities X, Y, X + Y 9, X + Y, then the point (, ) is the point that makes the objective function: R = X + Y in minimum value. 9

60 - If: sec (9 θ) = smallest positive angle., then θ =. where θ is the - If the ratio between the measures of the angles : :, then the radian measure of the smallest angle is rad and the radian measure of the greatest angle is rad. - ABC is an isosceles in which m (<A) =, then m (<B) =.. rad. - If A = 9, B =, then A + B =.. - sin sin cos = 7- If A = 8, then A T =., A T is of order. 8- If sin X = cos X then sin (X + Y) =.. 9- if cot (7 θ) = then θ = or θ = <θ<. -If θ is a positive angle in the standard position, its terminal side cuts the unit circle at the point (, ), then m (<θ) =.. -If Cosec (θ ) sec (θ ) = where < θ < 9 then sin θ =.. - If the arc length of a circle = its circumference, then the 8 central angle then subtends this arc its radian measure =.. - The angle whose measure is ( - 7 ) lies in quad. - If X Z + Z X =.., Y =.., Z =. - If sin (9 + θ) + = where 9 < θ < 8, then θ =. Y = 8

61 B- Answer: ) If A =, B = Find the matrix X S.T. X T = A T B T ) If A =, B = Find AB ) If A =, B = Find the matrix X S.T. AX = B. ) If A =, prove that A A + 9I = (A I) ) If A =, B = Find the matrix X such that X T A = B ) If A = ( - ), B=, C, Q = Prove that AB = QC T 7) If A =, B = 8 and C = 8) If A =, B = ( ) Find the matrix X such that X T = (AB) 9) If (A T B) T =, B = Find the matrix A.

62 C- Without using the calculator find the value of: ) Cos θ + tan ( θ 9 ) + cot (8 θ) sin (7 + θ) ) sin ( -) + sec + cos 8 ) If sin X = cos X where X [, π [ Find the value of sin X cosx ) If cos (9 + a) + = where π < a < π Find the value of cos (a 8 ) + cos ( ) cosec + tan. ) If sin a + = where 8 < a < 7, 8 tan b + = where 7 < b < and if cot (9a) cot ( b) 8 sec (8 b) tan (7 Find the value of X where X ], π [ = cot X a) ) Find the value of X such that tan X = sin sec + cos - where X ], π [. ) If < A complements < B and cos B = expression calculator. cot (9 A) tan B cosec(9 B) cosec (8 find the value of the A) without using the 7) If F (X) = sin X prove that F ( ) X F ( ) + cos π = 9 9 8) If tan a = where a is the measure of greatest positive angle where a ],π [ and 8 sec b 7= where b ] cot (7 a) tan( b) and cot C = 8sec (8 b) sec a Find C where < C <,π[

63 C- Find graphically S.S of the inequalities: ) X, X + Y <, X 8Y < ) X, Y, X <, Y < ) Y X, Y + 7X 8, Y + X D- A family buys two kinds of meat: the first 9% of pure meat and % of fats with the price L.E for each kilogram: while the second contains 7% of pure meat and % of fats with the price L.E for each kilogram. If the family needs to buy at least kg of pure meat and kg of fats weekly, find how much meat of the two kinds the family buys the least cost.