1 Week 1 Definition 1.1 Let a and b be numbers, a smaller than b. Then the set of all numbers between a and b : a and b included is denoted [a, b] a included, b excluded is denoted [a, b) a excluded, b included is denoted (a, b] a and b excluded is denoted (a, b) Proposition 1.1 The solutions of the equation ax + bx + c = 0 are : x + = b + a x = b a Where = b 4ac is called the determinant of the polynomial. Depending on whether it is negative, zero or positive, the equation has respectively zero, one or two solutions. Definition 1. A function from a set A to a set B is a rule that assigns to each element of A an element of B. Its graph is the set of pairs (a, b) such that: a A (is in A) and b = f(a) It can be plotted on a plane with x-y coordinates by setting (x, y) = (a, f(a)). Definition 1.3 A function is periodic is there exists a positive real number k such that : f(x + k) = f(x) for every x in the domain of f. The least such positive real number is called the period of the function. Definition 1.4 A rectangular coordinate system on the plane is a pair of perpendicular oriented lines with a common unit length. These lines are called axis and commonly denoted x and y. Definition 1.5 A function f is even if A function is odd if f( x) = f(x) f( x) = f(x) 1
Definition 1.6 For an angle θ, an initial side is a ray (half line) in the plane. The corresponding terminal side is the ray obtained by rotating the initial side by θ about its origin, counterclockwise for θ positive, clockwise for θ negative. terminal side vertex initial side Definition 1.7 Coterminal angles are angles that, given an initial side, share the same terminal side. Proposition 1. A full rotation is an angle of measure 360 degrees or π radians. Proposition 1.3 An arc of radius r and angle θ measures : s = rθ Proposition 1.4 A circular sector of radius r and angle θ has area : A = 1 r θ
Definition 1.8 Being given a rectangular coordinate system, the standard position is obtained by taking the positive x-axis as initial side. Definition 1.9 The unit circle is the circle of radius one centered on the origin. Definition 1.10 When put in standard position, an angle θ has its terminal side intersecting the unit circle. The assignment of the x-coordinate of the intersection point to the angle (in radians) defines a function called cosine. The assignment of its y- coordinate to the angle defines a function called sine. If the initial and terminal sides are not perpendicular, the latter intersects the x = 1 line. The assignment of the y- coordinate of the intersection point to the angle (expressed in radians) defines a third function called tangent. P sin t t cos t 1 tan t T Proposition 1.5 The similar triangle formula gives us : and the Pythagorean formula gives us : tanθ = sinθ cosθ cos θ + sin θ = 1 T 1 t cos t 1 P sin t tan t 3
Proposition 1.6 The sine and cosine functions for the angles π/6, π/4 and π/3 share the values : 1/, / and 3/. 3 3 4 1 6 1 3 Definition 1.11 Let θ not be a multiple of π, then its reference angle θ R is the acute angle that the terminal side of θ makes with the x-axis. Proposition 1.7 For angles θ (0, π ), θ R = θ θ ( π, π), θ R = π θ θ ( π, 3π ), θ R = θ π θ ( 3π, π), θ R = π θ Proposition 1.8 Week sin θ = sin θ R cos θ = cos θ R Proposition.1 Cosine and sine have period π. Tangent has period π. Proposition. Cosine is even, whereas sine and tangent are odd. Proposition.3 If y = a sin(bx + c) or y = a cos(bx + c), for a, b non-zero real numbers and c any real, then the graph has amplitude a, period π and phase shift c b. b 4
3 Quiz questions What is a function? What are periodic, even and odd functions? What are initial and terminal sides of an angle? What are coterminal angles? What is the standard position? How many radians is a degree? How many degrees is a radian? What is the reference angle of an angle? What are the relations relating the trigonometric functions of an angle with the ones of its reference angle? What is the unit circle? How are the sine, cosine and tangent functions defined? Give the trigonometric functions of the angles : 0, π 6, π 4, π 3, π. What are the period, amplitude and phase shift of the graph y = 10 sin( x 3 + 1 )? 4 Week 3 Proposition 4.1 Suppose an angle is in the standard position and M=(x,y) is on the terminal side. Then, defining r = x + y, we have : cos θ = x r sin θ = y x tan θ = y x There is a mnemotechnic method to remember it called SohCahToa, which means that, in the right triangle formed by O=(0,0), M=(x,y) and (x,0), sine is the ratio of the length of the opposite side over the one of the hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Proposition 4. When solving a right triangle, we have the following tools : Sum of angles The Pythagorean theorem SohCahToa Definition 4.1 The direction or bearing of a straight moving object is the acute angle that the trajectory makes with the north-south axis. 5
Definition 4. The triangle ABC is a convention where A,B and C are the three vertices, α, β and γ the associated angles, and a, b and c the lengths of the respective opposite sides. A c α b B β a γ C Proposition 4.3 In the triangle ABC, the Law of Sine holds : sin(α) a = sin(β) b = sin(δ) c Proposition 4.4 In the triangle ABC, the Law of Cosine holds : a = b + c bc cos(α) b = a + c ac cos(β) c = a + b ab cos(δ) 5 Week 4 Proposition 5.1 In the triangle ABC, the area A of the triangle is given by : A = s(s a)(s b)(s c) where s = a + b + c Proposition 5. When solving a general triangle, we have the following tools : Sum of angles The Cosine laws The Sine laws Definition 5.1 Let f be a function. Its inverse, if it exists, is the function, usually denoted f 1, satisfying : for every x in the domain of f. f(f 1 (x)) = x f 1 (f(x)) = x 6
Definition 5. Let f be a function. A left inverse, if it exists, is a function g satisfying : for every x in the domain of f. f(g(x)) = x Definition 5.3 The function cos 1 or arccos is defined by the following : Dom (cos 1 ) = [ 1, 1], Ran (cos 1 ) = [0, π] and cos 1 (x) = y if and only if x = cos(y) Definition 5.4 The function sin 1 or arcsin is defined by the following : [ Dom (sin 1 ) = [ 1, 1], Ran (sin 1 ) = π, π ] and sin 1 (x) = y if and only if x = sin(y) Definition 5.5 The function tan 1 or arctan is defined by the following : ( Dom (tan 1 ) = R, Ran (tan 1 ) = π, π ) and tan 1 (x) = y if and only if x = tan(y) Proposition 5.3 The inverse trig functions are not real inverses, they are left inverses, i.e. the following equalities hold : 6 Quiz What is a function? cos ( cos 1 (x) ) = x sin ( sin 1 (x) ) = x tan ( tan 1 (x) ) = x How are the sine, cosine and tangent functions defined? How do you find the sine, cosine and tangent values of an angle in a right triangle? What is the bearing of a moving boat? Give the sine law. Give the cosine laws. 7
What is an inverse function? What are te domains of arccos, arcsin and arctan? What is the range of arccos? What is the range of arcsin? What is the range of arctan? What equations link trigonometric functions to their inverses? 7 Week 5 Proposition 7.1 The following identities are called the sum formulas : cos(u + v) = cos(u)cos(v) sin(u)sin(v) sin(u + v) = sin(u)cos(v) + cos(u)sin(v) Proposition 7. From the sum formulas can be recovered the following relations, using symmetry properties : cos(u v) = cos(u)cos(v) + sin(u)sin(v) cos(u) = cos (u) sin (v) sin(u v) = sin(u)cos(v) cos(u)sin(v) sin(u) = sin(u)cos(u) tan(u) + tan(v) tan(u + v) = 1 tan(u)tan(v) tan(u) tan(v) tan(u v) = 1 +tan(u)tan(v) tab(u) = tan(u) 1 tan (u) Proposition 7.3 And from the double-angle formula for cosine derived above can be found the half-angle formulas : sin ( u ) = 1 cos(u) cos ( u ) = 1 + cos(u) tan ( u ) = 1 cos(u) 1 + cos(u) 8
Proposition 7.4 In the case of tangent, we can determine that : tan ( u ) = sin(u) 1 + cos(u) tan ( u ) = 1 cos(u) sin(u) Proposition 7.5 The following equalities hold : sin(u)cos(v) = 1 ( ) sin(u + v) + sin(u v) cos(u)cos(v) = 1 ( ) cos(u + v) + cos(u v) sin(u)cos(v) = 1 ( ) cos(u + v) cos(u v) Proposition 7.6 The following equalities hold : 8 Week 6 sin(u) + sin(v) = sin( u + v cos(u) + cos(v) = cos( u + v cos(u) cos(v) = sin( u + v )cos( u v ) )sin( u v ) )sin( u v ) Remark: Being given a cartesian coordinate system (x-y axis), it is often understood that the polar coordinate system is the one obtained by taking the positive x-axis as a reference axis. Proposition 8.1 In the standard coordinate systems, the equations relating both systems are : x = r cos(θ) y = r sin(θ) and r = x + y tan(θ) = y/x if x 0 Definition 8.1 Let i be an object such that i = 1, then we can define the complex numbers as the set of numbers : z = a + ib where a, b R The addition and multiplication of complex numbers are given by (a + ib) + (c + id) = (a + c) + i(b + d) (a + ib)(c + id) = ac + iad + ibc + i bd = (ac bd) + i(ad + bc) 9
Complex numbers being esentially a pair of real numbers, they can be plotted on a plane equipped with a coordinate frame. The distance from the corresponding point to the origin is called the norm of the number. Definition 8. The absolute value or norm of a complex number z = a + ib is : z = a + b ( ) a The complex number z can then be rewritten z = z z + i b z z = 1, i.e. z is on the unit circle. = z z, with Proposition 8. Any complex number z can be written the following way : z = r ( cos(θ) + i sin(θ) ) := r cis(θ) where r is its norm and θ its argument. Proposition 8.3 If z 1 = r 1 cis(θ 1 ) and z = r cis(θ ), then : z 1 z = r 1 r cis(θ 1 + θ ) z 1 = r 1 cis( θ 1 ) Proposition 8.4 (De Moivre s) Let z C,then : Proposition 8.5 Let z C,then : z n = z n cis(nθ z ) z 1 1 θ z + kπ n = z n cis( ) for k {0, 1,,..., n 1} n Definition 8.3 As angles caracterised a rotation, vectors caracterise translation. Therefore a vector is a displacement in the plane. It associates to any point in the plane a new, accordingly displaced point. Proposition 8.6 In a cartesian coordinate system, a vector is caracterised by a pair of numbers : displacement in one direction, displacement in the other. In a polar coordinate system, a vector is caracterised by a pair of numbers too, its amplitude (or length) and direction (an angle in [0, π)). Proposition 8.7 Vectors can be added together and multiplied by a real number. In cartesian coord. these operations are given by : (u, v) + (x, y) = (u + x, v + y) a(u, v) = (au, av) Definition 8.4 Let U = (u 1, u ) and V = (v 1, v ) be vectors, then their dot product is a real number given by : U.V = u 1 v 1 + u v 10
9 Quiz 3 Give the sum formulas. Derive the difference formulas. Give the double-angle formulas. Give the half angle formulas. What is the norm of a complex number? number using trigonometric functions? And how do you write a complex Give the formulas for multiplication of complex numbers, both in usual and trigonometric form. Give the DeMoivre s formula. Give the nth root formula. What is a vector? Define addition of vectors. Define the dot product of vectors. 10 Week 7 Definition 10.1 A line in the plane with coordinate frame can be described by the graph of a function of the form : f(x) = mx + p The number m is called the slope and p the y-intercept. Proposition 10.1 The line passing through the points (a, b) and (c, d) has slope : m = a c b d Proposition 10. The line of slope m passing through the point (a, b) has equation given by : y b = m(x a) Proposition 10.3 Let two lines have respective slope m and n. Then : the lines are parallel if and only if n = m the lines are perpendicular if and only if nm = 1 11
Definition 10. A parabola is the set of all points equidistant from a given point F (for focus) and a given line l (called directrix)in the plane. Proposition 10.4 In the proper coordinate system (u,v), a parabola is given by the solutions of : v = 1 4p u In this setting, the vertex is at (0,0), the focus at (0,p) and the directrix is given by v=-p. v p F p V directrix u Definition 10.3 An ellipse is the set of all points in a plane, the sum of whose distance to two given points in the plane (the foci) is a positive constant. Proposition 10.5 In the proper coordinate system (u,v), a ellipse is given by the solutions of : ( u ) ( v ) + = 1 with a > b > 0 a b In this setting, the distance in between u and v-intercepts are respectively a and b and the foci are at c = a b from the origin. v F b F c a 1 u 1
Definition 10.4 The eccentricity e of an ellipse is the ratio of the distance from center to focus by the distance from center to vertex, i.e.: e = c a Definition 10.5 An hyperbola is the set of all points in a plane, the difference of whose distance to two given points in the plane (the foci) is a positive constant. Proposition 10.6 In the proper coordinate system (u,v), a ellipse is given by the solutions of : ( u a ) ( v b ) = 1 with a > b > 0 In this setting, the distance in between u-intercepts are a and the foci are at c = a + b from the origin. v F a F 1 a b u Definition 10.6 Being given a half-line in the plane, we can give the position of any point P in the plane by specifying its distance r P from the origin O of this half-line and the angle θ P it makes with OP. The assignment (r P, θ P ) to P makes a polar coordinate system on the plane. 11 week 8 Proposition 11.1 In polar form, equations of conics (parabolas, ellipses, hyperbolas) have the following form : where d > 0 r = de 1 ± e cosθ or r = de 1 ± e sinθ If e = 1 the graph is a parabola, if e (0, 1) an ellipse and if e > 1 an hyperbola. 13
1 Final Quiz Definition of a function and of its graph. Definition of cosine, sine and tangent using the unit circle. Give the conversion radiants-degree and vice-versa. Define reference angles and tell how its trig functions relate to the original angle. Define cosecant, secant and cotangent. Give the symetries of the sine, cosine and tangent functions. Give the graphs of the sine, cosine and tangent functions. Give the domain and range of inverse cosine, inverse sine and inverse tangent. Give the relationship between trig function and their inverses. Give the Pythagorean theorem applied to the unit circle. Give the SohCahToa formulas. Give the sum of angles formulas. Know how to calculate arc lengths. Give sine rule. Give cosine rules. Know how to find the area of a triangle. Write a complex number in two different ways. Multiply two complex numbers and give the inverse of a complex number. Give the nth root formula. Give the general equations for ellipses, hyperbolas and parabolas. 14