Algebra Matrices: • A matrix of order m × n , m: number of rows , n: number of columns • Multiplying a real number by a matrix means multiplying each element of the elements of the matrix by that real number • 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 • Multiplying matrices : 𝐴 𝑚 × 𝑛 , 𝐵 𝑛 × 𝑓 , then ( 𝐴 × 𝐵 ) 𝑚 × 𝑓 • (AB) t = B t A t • Determinant: if a square matrix A = ( 𝑎 𝑏 𝑐 𝑑 ) , • Determinant of the triangular matrix: • A matrix of order n × n , k ∈ ℝ , then | 𝑘𝐴 | = k n | 𝐴 | • Finding the area of a triangle by using determinants • Solving a system of linear equations by C ramer's rule : ax + by = m , cx + dy = n • And the same for the system equation of three variables x , y and z. • Multiplicative inverse of a matrix : A = ( 𝑎 𝑏 𝑐 𝑑 ) , A - 1 = 1 | 𝑎 𝑏 𝑐 𝑑 | ( 𝑑 − 𝑏 − 𝑐 𝑎 ) توجيه الرياضيات مفاهيم الرياضيات الصف الأ ول الثانوي الفصل الدراسي الثاني (لغات) ( 1 ) Properties of determinants: ✓ | 𝐴 | = | 𝐴 𝑡 | ✓ The value of the determinant equals zero if - If all the elements of any row (column) equal zero - If all the corresponding elements of any rows (columns) are equal. ✓ If there is a common factor in all elements of any row (column) , taken outside the ✓ If the position of two rows (columns) are interchanged, then the resulting determinant = - the original determinant .............................................................................................. Trigonometry Basic trigonometric identities: ✓ sec θ = 1 𝑐𝑜𝑠 θ , csc θ = 1 𝑠𝑖𝑛 θ , cot θ = 1 𝑡𝑎𝑛 θ = 𝑐𝑜𝑠 θ 𝑠𝑖𝑛 θ ✓ sin 2 θ + cos 2 θ = 1 ✓ tan 2 θ + 1 = sec 2 θ ✓ cot 2 θ + 1 = csc 2 θ The general solution of the trigonometric equation: If 𝛽 the smallest positive measure satisfies the equation , n ∈ ℤ , then: ✓ The general solution of the equation sin θ = a is θ = 𝛽 + 2 𝜋 n , θ = ( 𝜋 − 𝛽 ) + 2 𝜋 n ✓ The general solution of the equation cos θ = a is θ = ± 𝛽 + 2 𝜋 n ✓ The general solution of the equation tan θ = a is θ = 𝛽 + 𝜋 n • The area of circular sector = 𝑥 𝑜 360 𝑜 × 𝜋 𝑟 2 = 1 2 θ 𝑟𝑎𝑑 𝑟 2 = 1 2 l r • The perimeter of the circular sector = 2 r + l • The area of circular segment = 1 2 𝑟 2 ( θ 𝑟𝑎𝑑 - sin θ ) • The area of the triangle = 1 2 the product of the lengths of two sides × sine of the included angle between them. • The area of the triangle = √ 𝑆 ( 𝑆 − 𝑆 1 ) ( 𝑆 − 𝑆 2 ) ( 𝑆 − 𝑆 3 ) , S is half perimeter of the triangle , 𝑆 1 , 𝑆 2 , 𝑆 3 are the length of its sides. • The area of the quadrilateral = 1 2 the product of the lengths of its diagonals × sine of the included angle between them. • The area of the regular polygon = 1 4 𝑛 𝑥 2 cot 𝜋 𝑛 , n: number of its sides, x: length of its side ( 2 ) Geometry • The norm of the vector 𝐴 ⃑ (x,y) = ‖ 𝐴 ⃑ ‖ = √ 𝑥 2 + 𝑦 2 • The polar form of the vector 𝐴 ⃑ = ( ‖ 𝐴 ⃑ ‖ , θ ) • The fundamental form of the vector 𝐴 ⃑ (x,y) = x 𝑖 ⃑ + y 𝑗 ⃑ • • Operation on vectors: • Divides directed line segment: • • The different forms of the equation of straight line: Internally Externally ( 3 ) ( 4 )