Pre-Calculus PDF

Pre-Calculus Made by MATRIX (those who don’t know 💀💀💀) Table of Contents Table of Contents 0 Supplementary Knowledge 1 Conic Sections 2 Circle 3 Examples...

Pre-Calculus Made by MATRIX (those who don’t know 💀💀💀) Table of Contents Table of Contents 0 Supplementary Knowledge 1 Conic Sections 2 Circle 3 Examples 3 Parabola 4 Examples 4 Ellipse 5 Examples 5 Hyperbola 6 Examples 6 Systems of Nonlinear Equations 7 Linear vs Nonlinear 7 Solutions 7 Substitution Method 7 Elimination Method 8 Sequence, Series, & Sigma Notation 9 Sigma Notation 9 Examples 9 Mathematical Induction 10 Steps 10 Examples 10 Binomial Theorem 11 Pascal’s Triangle 11 Properties of Binomial Theorem 11 Examples 11 Finding the nth Term 12 Examples 12 Circular Functions 13 Angles in Standard Position 13 Quadrantal Angle 13 Trigonometric Functions 14 Left Hand Trick 15 Reference Angles 15  1 Supplementary Knowledge If you don’t know this, you’re cooked ggs spatchcocked those who know Midsegment Theorem  2 Conic Sections Section Description Picture Circle The plane parallel to the bases crosses one of the cones. Ellipse The plane is tilted at an angle. Parabola The plane is tilted further and becomes parallel to a cone’s surface. Hyperbola The plane is tilted further so that it intersects both cones.  3 Circle A circle is a set of all the points on a plane that are equidistant from a fixed point. This fixed point is called the center and the fixed point distance is called the radius. Examples  4 Parabola A parabola is the set of all points on a plane equidistant from a fixed point (focus) and a fixed line (directrix). Examples  5 Ellipse Collection of all points in a plane such that the sum of distances of each point from fixed points in the plane is constant Examples  6 Hyperbola Is the set of points in a plane such that the differences in distances of each from two fixed points in the plane is constant Examples Ima add later  7 Systems of Nonlinear Equations These are systems of two or more equations in two or more variables containing at least one equation that is not linear. Linear vs Nonlinear 1. Linear Equation ○ Forms a straight line on the graph ○ Equation has the maximum degree of 1 2. Nonlinear Equation ○ Forms a curve on the graph ○ Equation has a maximum degree of 2 or more Solutions Substitution Method The substitution method is one of two ways to solve systems of nonlinear equations. In this example, (𝑥 + 𝑦) in the first equation was substituted with 7 from the 2nd equation.  8 Elimination Method The elimination method is another way to solve systems of nonlinear equations. In 2 this example, the second equation was multiplied by -1 so that 𝑥 can be canceled when the two equations are added.  9 Sequence, Series, & Sigma Notation A sequence is a list of numbers, stated in a particular order, such that each number can be obtained according to a certain rule. Meanwhile, a series represents the sum of the terms of the sequence. Sigma Notation In mathematics, we use sigma notation (aka summation notation) to denote a sum. Examples  10 Mathematical Induction Mathematical induction is a technique of proving a statement, theorem or formula which is thought to be true, for each and every natural number n. Steps 1. Verification ○ The statement is verified for 𝑛 = 1. 2. Induction Hypothesis ○ Assume that the statement is true for 𝑛 = 𝑘. 3. Prove ○ Prove that the statement is true for 𝑛 = 𝑘 + 1. 4. Conclusion ○ Since the statement is true for 𝑛 = 1, and it’s true for 𝑛 = 𝑘 + 1, provided that it’s true for 𝑛 = 𝑘, thus, it’s true for any natural number (𝑛). Examples  11 Binomial Theorem A binomial is an expression that has two terms. Binomial expansion is an 𝑛 expansion of the form (𝑎 + 𝑏) , expressed as a polynomial. The binomial theorem is a theorem that can help you expand a binomial raised to a specific power. Pascal’s Triangle Pascal’s triangle is closely related to the binomial theorem. The first row consists of 1. Each row after the first consists of one term more than the previous row. Each row after the first begins and ends with 1. Every value, other than the first and the last, can be obtained by adding the pair of numbers immediately above it. The numbers in each row are symmetric with respect to the center. Properties of Binomial Theorem The exponent of a starts from n, decreases by 1 each time, and equals 0 in the last term The exponent of b starts at 0 and increases by 1 each time until it reaches n in the last term The coefficients of the terms are the numbers in the Pascal's Triangle that correspond to n Examples  12 Finding the 𝑛𝑡ℎ Term The formula to find the nth term is the same as the binomial theorem but without the sigma. Examples  13 Circular Functions Angle (Plane Geometry) In plane geometry, an angle is a union of two non-collinear rays with a common endpoint Angle (Trigonometry) It is the amount of rotation that the terminal side makes with respect to the initial side Initial Side: if the position is stationary Terminal Side: if the position is moving Angles in Standard Position If its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis. Quadrantal Angle An angle is quadrantal if the terminal side coincides with the x-axis or y-axis. Its measurements are always multiples of 90 degrees. Examples include 90°, 180°, 270°, etc…  14 Co-terminal Angles Angles that are in standard position which have the same terminal side. Trigonometric Functions Circular Functions  15 Left Hand Trick Reference Angles