Pre-Calculus Course Study Guide

Pre-Calculus: Course Study Guide ❖ Prerequisites: Fundamental Concepts of Algebra Algebraic Expressions and Real Numbers ● Natural Numbers: positive numbers ● Whole Numbers: number without fractions ● Integers: whole numbers and their opposite ● Simple Fraction: integer/natural ● Absolute Value: distance from 0 ● Evaluate -> Expressions ○ 2x+3; when x=3 ● Solve -> Equations ○ 11= 2x+3 ● Intersection (∩): in both brackets at the same time ○ Ex: 𝐴 = {1,3,4}𝐵 = {3,4,5,6}= A∩B: {3,4} ● Union (∪): joining of things, join what’s inside ● { }: empty set; ⊘: “null set” = no values Radicals ● Radical: the main/primary root of what’s underneath ○ √36 = square root 3 ○ √8 = cube root ■ √4𝑥 2 = 2|𝑥| ● the absolute value keeps the x from becoming negative ● IMPORTANT: The only time you need the absolute value is when you take an even root of an even power and get an odd result. (EEO) ○ Examples ■ √49𝑥 2 𝑦 4 =7|𝑥3 |𝑦2 3 3 3 ■ 2 √5+ √5= 3 √5 Factoring Formulas ● (a+b)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 ○ 49𝑥 2 + 126𝑥 + 81 = (7𝑥 + 9)2 ● (a+b)(a-b) = 𝑎2 − 𝑏 2 (difference of 2 squares) ● 𝑎2 + 𝑏 2 = sum of 2 squares (can’t factor) = prime ● 𝑎3 − 𝑏 3 = (𝑎 − 𝑏) (𝑎2 + 𝑎𝑏 + 𝑏 2 )= can never factor further than this! ● 𝑎3 + 𝑏 3 =sum of 2 cubes ○ (a+b) (𝑎2 − 𝑎𝑏 + 𝑏 2 ) Rational Expressions ● Simplifying Rational Expressions ○ 1) Factor the numerator and denominator completely ○ 2) Divide both the numerator and denominator by any common factors ■ Ex. 𝑥 3 +𝑥 2 𝑥 2 (𝑥+1) 𝑥+1 = 𝑥+1 = 𝑥 2 (𝑥+1) (𝑥+1) - cancel (x+1) ● = 𝑥 2 , 𝑥 ≠ −1 ● Multiplying Rational Expressions ○ 1) Factor all numerators and denominators completely ○ 2) Divide numerators and denominators by common factors ○ 3) Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators ● Dividing Rational Expressions ○ 1) Find the answer by inverting the second divisor and multiplying using the steps above ● Adding/Subtracting Rational Expressions That Have Different Denominators ○ 1) Find the LCD of the rational expression ○ 2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. ○ 3) Add/Subtract numerators by placing the resulting expression over the LCD. ○ 4) If possible, simplify. Finding Domain ● Ex. (𝑥−3)(2𝑥+1) (𝑥+1)(𝑥−3) = 2𝑥+1 𝑥+1 =Domain = (-∞, −1) ∪ (−1, ∞) Equations ● Linear Equation: an equation that is written in the form ax+b=0 where a and b are real numbers and a≠ 0 ● Solving Linear Equation ○ 1) Simplify both expressions ○ 2) Group all variable terms on one side and constant terms on the other ○ 3) Isolate the variable and solve ● Solving Rational Equations ○ 1) Find the least common denominator ○ 2) Use the distributive property and divide out common factors ○ 3) Simplify ● Solving Equation Involving Absolute Value ○ Isolate absolute value ○ Split into 2 different equations ○ Solve for both ● Square Root Property ○ If 𝑢2 = 𝑑, 𝑡ℎ𝑒𝑛 𝑢 = √𝑑 𝑜𝑟 𝑢 = −√𝑑 ● Solving Radicals Containing nth Roots ○ 1) Arrange terms so one radical is isolated on one side of the equation. ○ 2) Raise both sides of the equation to the nth power to eliminate the nth root. ○ 3) Solve the resulting equation. ○ Check all solutions in the original equation. ■ Ex. 𝑆𝑜𝑙𝑣𝑒: √2𝑥 − 1 + 2 = 𝑥 ■ Step 1: √2𝑥 − 1 + 2 = 𝑥 → √2𝑥 − 1 = 𝑥 − 2 ■ Step 2: (√2𝑥 − 1 ) 2 = (𝑥 − 2) 2 → 2𝑥 − 1 = 𝑥 2 − 4𝑥 + 4 ■ Step 3: 2𝑥 − 1 = 𝑥 2 − 4𝑥 + 4 ■ 0 = 𝑥 2 − 6𝑥 + 5 → 0 = (𝑥 − 1)(𝑥 − 5) → 𝑥 = 1 𝑜𝑟 𝑥 = 5 Linear Inequalities and Absolute Value Inequalities ● Interval Notation: represents subsets of real numbers ○ Open Interval: (a.b) represents the set of real numbers between, but not including a and b ○ Closed Interval: [𝑎, 𝑏]represents the set of real numbers between, and including a and b ○ Infinite Interval: (𝑎, ∞)represents the set of numbers that are greater than a ○ Infinite Interval: (−∞, 𝑏] represents the set of real numbers that are less than or equal to b ● Graphing Intersections and Unions ○ 1) Graph each interval on a number line ○ 2A) To find the intersection, find the set of numbers on the number line where both graphs have the set in common ○ 2B) To find the union, take the portion of the number line representing the total collection of numbers in the two graphs ● Solving an Absolute Value Inequality ○ If X is an algebraic expression and c is a positive number ■ 1) The solutions of |𝑋|< c are the numbers that satisfy -c < X 𝑐 are the numbers that satisfy X < -c or X > c ❖ Graphs and Functions Graphs and Graphing Utilities ● X- axis: horizontal number line ● Y- axis: vertical number line ○ Each point corresponds to an ordered pair (x,y) Basics of Functions and Their Graphs ● Relation: any set of ordered pairs ○ Set of first components is the domain (to find the domain, look for all inputs on the x-axis that correspond to the points on the graph). The set of second components is the range (to find the range, look for all the outputs on the y-axis that correspond to points on the graph ● Vertical Line Test: if any vertical line intersects the graph in more than one point, the graph is not a function ● Zeros of a function: values of x for which f(x) = 0 ○ A function can have more than one x-intercept, but at most one y-intercept ● Difference Quotient: 𝑓 (𝑥+ℎ)−𝑓(𝑥) ℎ ,ℎ = 0 ● Piecewise Function: defined by two (or more) equations over a specified domain ● Relative Maximum: the “peak” of the graph ● Relative Minimum: the “bottom” of the graph ● Even Function: f(-x) = f(x); symmetric to the y-axis ● Odd Function: f(-x) = -f(x); symmetric to the origin Linear Functions and Slope 𝑦 2 −𝑦 1 ● To find slope: 2 𝑥 −𝑥 1 ● Point Slope: y-𝑦1 = 𝑚(𝑥 − 𝑥1 ) ● General Form: Ax + By + C = 0 ● Average Rate of Change: 𝑦 𝛥 𝛥𝑥 = 𝑓(𝑥2 )−𝑓(𝑥) 𝑥2 −𝑥1 Transformations of Functions ● Vertical Shifts ○ The graph of 𝑦 = 𝑓(𝑥) + 𝑐is the graph of 𝑦 = 𝑓(𝑥)shifted c units up. ○ The graph of 𝑦 = 𝑓(𝑥) − 𝑐is the graph of 𝑦 = 𝑓(𝑥)shifted c units down. ● Horizontal Shifts ○ The graph of 𝑦 = 𝑓(𝑥 + 𝑐)is the graph of 𝑦 = 𝑓(𝑥)shifted left c units. ○ The graph of 𝑦 = 𝑓(𝑥 − 𝑐)is the graph of 𝑦 = 𝑓(𝑥)shifted right c units. ● Vertically Stretching and Shrinking Graphs ○ In the graph 𝑦 = 𝑐𝑓(𝑥), if 𝑐 > 1the graph of 𝑦 = 𝑓(𝑥)is vertically stretched by multiplying each of its ycoordinates by c. ○ In the graph 𝑦 = 𝑐𝑓(𝑥), if 0 < 𝑐 < 1, the graph 𝑦 = 𝑓(𝑥) is vertically shrunk by multiplying each of its ycoordinates by c. ● Horizontally Stretching and Shrinking ○ In the graph 𝑦 = 𝑓(𝑐𝑥), if 𝑐 > 1the graph of 𝑦 = 𝑓(𝑥)is horizontally shrunk by dividing each of its x-coordinates by c. ○ In the graph of 𝑦 = 𝑓(𝑐𝑥), if 0 < 𝑐 < 1, the graph of 𝑦 = 𝑓(𝑥)is horizontally stretched by diving each of its xcoordinates by c. Inverse Functions ● f(g(x)) = x and g(f(x)) = x, the function g is the inverse of function f which means f and g are inverses of each other ● Finding a Function’s Inverse ○ Switch x and y, then solve for y ● Horizontal Line Test: If function 𝑓 has an inverse that is a function 𝑓 −1 , if there is no horizontal line that intersects the graph of the function 𝑓at more than one point. Distance and Midpoint Formulas: Circles ● Distance Formula: 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 ● Midpoint Formula: ( 𝑥1 +𝑥2 𝑦1 +𝑦2 2 , 2 ) ● Standard Form of Circle: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 ● General Form of Circle Equation: 𝑥 2 + 𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 ❖ Polynomial and Rational Functions Complex Numbers ● Imaginary Unit 𝑖is defined as 𝑖 = √−1, 𝑤ℎ𝑒𝑟𝑒 𝑖 2 = −1 ● In 𝑎 + 𝑏𝑖, 𝑎 is the real part and 𝑏is the imaginary part ● Complex Numbers: real numbers a and b, and i, the imaginary unit ● Pure Imaginary Number: imaginary number in the form of 𝑏𝑖 ● Adding and Subtracting Complex Numbers ○ (𝑎 + 𝑏𝑖) + (𝑐 + 𝑑𝑖) = (𝑎 + 𝑐) + (𝑏 + 𝑑) 𝑖 ■ Add complex numbers by adding the real parts, adding the imaginary parts, and expressing the sum as a complex number ○ (𝑎 + 𝑏𝑖) − (𝑐 + 𝑑𝑖) = (𝑎 − 𝑐) + (𝑏 − 𝑑) 𝑖 ■ Subtract complex numbers by subtracting the real parts, subtracting the imaginary parts, and expressing the sum as a complex number ● Multiplying and Dividing Complex Numbers ○ For multiplication, FOIL, first, outer, inner, last, then solve. ○ For division, multiply the numerator and the denominator by the complex conjugate of the denominator. Use the FOIL method. Combine any imaginary terms, then combine real terms. ● Complex Conjugate of the number 𝑎 + 𝑏𝑖is 𝑎 − 𝑏𝑖 ● Principle Square Root: for any positive real number 𝑏, the principal square root of the negative number −𝑏is defined by √−𝑏 = 𝑖√𝑏 Quadratic Functions ● Quadratic Function is of the form: 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, 𝑎 ≠ 0 ● Standard Form of Quadratic Function: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘, 𝑎 ≠ 0 ○ Vertex: (h,k) ● Maximum and Minimum of Quadratic Functions ○ Function 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ■ If 𝑎 > 0, then f has a minimum that occurs at 𝑥 value 𝑓(− 𝑏 2𝑎 = − 𝑏 2𝑎 2𝑎 , minimum ) ■ If 𝑎 < 0, then f has the maximum that occurs at 𝑥 = − value 𝑓(− 𝑏 𝑏 , maximum 2𝑎 ) Polynomial Functions and Their Graphs ● Polynomial Function: a function comprising more than one power function where the coefficients are assumed to not equal zero. The term with the highest degree is the leading term. ● Even multiplicity: the graph touches the x-axis and turns around at r ● Odd multiplicity: the graph crosses the x-axis at r ● Graphing a Polynomial Function ○ 1) Use the Leading Coefficient Test to determine graph’s end behavior ○ 2) Find x-intercept by setting f(x) = 0 and solving the resulting polynomial equation. ○ 3) Find the y-intercept by computing f(0) ○ 4) Use symmetry, if applicable to help draw the graph. ■ Y-axis symmetry: f(-x) = f(x) ■ Origin symmetry: f(-x) = -f(x) ○ 5) Use the fact that the maximum number of turning points of the graph is n-1 to check whether its drawn correctly ● Intermediate Value Theorem: If f is a polynomial function and f(a) and f(b) have opposite signs, there is at least one value of c between a and b for which f(c) = 0. Dividing Polynomials ● Division Algorithm ○ 𝑓(𝑥) = 𝑑(𝑥) ∗ 𝑞(𝑥) + 𝑟(𝑥) ○ f(x) = dividend ○ d(x) = divisor ○ q(x) = quotient ○ r(x) = remainder ● Long Division of Polynomials ○ 1) Set up long division ○ 2) Divide the 1st term of the dividend with the divisor ○ 3) Multiply by the divisor ○ 4) Write the answer and subtract ○ 5) Bring down the next number to the right ○ 6) Repeat Step 2 ○ 7) Write final answer ● Remainder Theorem ○ If the polynomial f(x) is divided by x-c, then the remainder is f(c). ● Factor Theorem ○ Let f(x) be a polynomial ■ a) If f(c) = 0, then x-c is a factor of f(x) ■ b) If x-c is a factor of f(x), then f(c) = 0 Zeros of Polynomial Functions ● Rational Zero Theorem ● Number of Roots ○ If f(x) is a polynomial degree n ≥1, counting multiple roots separately, the equation f(x) = 0 has n roots. ● Descartes’s Rule of Signs ○ The number of positive real zeros of f equals the number of sign changes of f(x) or is less than that number by an even integer. The number of negative real zeros of f applies a similar statement to f(-x). Rational Functions and Their Graphs ● Arrow Notation ○ 𝑥 → 𝑎+ : x approaches a from the right ○ 𝑥 → 𝑎− : x approaches a from the left ○ 𝑥 → ∞: x approaches infinity ○ 𝑥 → −∞: x approaches negative infinity ● Vertical Asymptote ○ If f(x) increases or decreases without bond as x approaches a ● Horizontal Asymptote ○ If f(x) approaches b as x increases or decreases without bond ● Graphing Rational Functions 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) ○ Determine whether the graph has symmetry ■ f(-x) = f(x): y-axis symmetry ■ f(-x) = -f(x): origin symmetry ○ Find the y-intercept by evaluating f(0) ○ Find the x-intercept by solving the equation p(x)=0 ○ Find any vertical asymptotes by solving the equation q(x) = 0 ○ Find the horizontal asymptote by using the rule for determining the horizontal asymptote of a rational function. ○ Plot at least one point between and beyond each x-intercept and vertical asymptote. ○ Use info above to graph the function between the asymptotes ❖ Exponential and Logarithmic Functions Exponential Functions ● Parent Function: 𝑓(𝑥) = 𝑏 𝑥 , where b is base and b > 0, 𝑏 ≠ 1 ● Natural exponential function: 𝑓(𝑥) = 𝑒 𝑥 1 ● Irrational number (natural base): 𝑒 ≈ 2.7183, 𝑒is the value that (1 + 𝑛)2 Logarithmic Functions ● Logarithmic Functions: for x>0 and b>0, b≠ 1 Exponential and Logarithmic Equations ● Exponential equation: equation containing a variable in an exponent ● Solving Exponential Equations by Expressing Each Side as a Power of the Same Base ○ If 𝑏 𝑀 = 𝑏 𝑁 , 𝑡ℎ𝑒𝑛 𝑀 = 𝑁 ● Using Natural Logarithms to Solve Exponential Equations ○ 1) Isolate the exponential expression ○ 2) Take the natural logarithm on both sides of the equation ○ 3) Simplify using one of the following ■ 𝐼𝑛 (𝑏 𝑥 ) = 𝑥 𝑙𝑛 (𝑏) 𝑜𝑟 𝑙𝑛 𝑒 𝑥 = 𝑥 ○ 4) Solve for the variable ● Using the Definition of a Logarithm to Solve Logarithmic Equations ○ 1) Express the equation in the form 𝑙𝑜𝑔𝑏 𝑀 = 𝑐. ○ 2) Use the definition to rewrite the equation in exponential form 𝑙𝑜𝑔𝑏 𝑀 = 𝑐 𝑚𝑒𝑎𝑛𝑠 𝑏 𝑐 = 𝑚 ○ 3) Solve for the variable. ❖ Trigonometric Functions Angles and Radian Measure ● Angle: two rays with a common endpoint called the vertex ● Quadrantal Angle: angle with its terminal side on the x-axis or the y-axis 𝑠 ● Radian Measure: 𝜃 = 𝑟 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ● To convert from degrees to radians, multiply by 𝜋𝑟𝑎𝑑𝑖𝑎𝑛𝑠 180∘ ● To convert from radians to degrees, multiply by 180∘ 𝜋𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ● Coterminal Angles: angles with the same initial and terminal sides ● Length of a Circular Arc: 𝑠 = 𝑟𝜃 The Unit Circle ● Definitions of the Trigonometric Functions in Terms of a Unit Circle ○ If 𝑡 is a real number and 𝑃 = (𝑥, 𝑦) is a point on the unit circle that corresponds to 𝑡, then 𝑠𝑖𝑛 𝑡 = 𝑦 𝑐𝑠𝑐 𝑡 = 1 ,𝑦 ≠ 0 𝑦 𝑐𝑜𝑠 𝑡 = 𝑥 𝑠𝑒𝑐 𝑡 = 1 ,𝑥 ≠ 0 𝑥 𝑡𝑎𝑛 𝑡 = 𝑦 ,𝑥 ≠ 0 𝑥 𝑐𝑜𝑡 𝑡 = 𝑥 ,𝑦 ≠ 0 𝑦 ● Domain and Range of Sine and Cosine Functions ○ Domain to sine and cosine function (−∞, ∞), set of all real numbers ○ Range [−1,1] ● Even and Odd Trigonometric Functions ○ Even ■ 𝑐𝑜𝑠 (−𝑡) = 𝑐𝑜𝑠 𝑡 𝑎𝑛𝑑 𝑠𝑒𝑐(−𝑡) = 𝑠𝑒𝑐 𝑡 ○ Odd ■ 𝑠𝑖𝑛(−𝑡) = −𝑠𝑖𝑛 𝑡, 𝑡𝑎𝑛(−𝑡) = −𝑡𝑎𝑛(𝑡), 𝑐𝑠𝑐(−𝑡) = −𝑐𝑠𝑐 𝑡, 𝑐𝑜𝑡(−𝑡) = − 𝑐𝑜𝑡(𝑡) ● Quotient Identities 𝑠𝑖𝑛 𝑡 ○ 𝑡𝑎𝑛 𝑡 = 𝑐𝑜𝑠 𝑡 𝑎𝑛𝑑 𝑐𝑜𝑡 𝑡 = 𝑐𝑜𝑠 𝑡 𝑠𝑖𝑛 𝑡 ● Pythagorean Identities ○ 𝑠𝑖𝑛2 + 𝑐𝑜𝑠 2 𝑡 = 1 1 + 𝑡𝑎𝑛2 𝑡 = 𝑠𝑒𝑐 2 𝑡 1 + 𝑐𝑜𝑡 2 𝑡 = 𝑐𝑠𝑐 2 𝑡 ● Periodic Functions: a function that repeats its values at regular intervals ○ Periodic Properties for Sine and Cosine (period 2𝜋) ■ 𝑠𝑖𝑛 (𝑡 + 2𝜋) = 𝑠𝑖𝑛 𝑡 𝑎𝑛𝑑 𝑐𝑜𝑠 (𝑡 + 2𝜋) = 𝑐𝑜𝑠 𝑡 ○ Periodic Properties for Tangent and Cotangent (period 𝜋) ■ 𝑡𝑎𝑛 (𝑡 + 𝜋) = 𝑡𝑎𝑛 𝑡 𝑎𝑛𝑑 𝑐𝑜𝑡 (𝑡 + 𝜋) = 𝑐𝑜𝑡 𝑡 Right Triangle Trig ● Right Triangle Definitions for Trig Functions ○ 𝑠𝑖𝑛 𝜃 = 𝑎 𝑐 𝑐𝑠𝑐 𝜃 = 𝑐 𝑎 𝑏 𝑐𝑜𝑠 𝜃 = 𝑐 𝑐 𝑠𝑒𝑐 𝜃 = 𝑏 𝑡𝑎𝑛 𝜃 = 𝑎 𝑏 𝑐𝑜𝑡 𝜃 = 𝑏 𝑎 ● Cofunction Identities 𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠 (90° − 𝜃) 𝑐𝑜𝑠 𝜃 = 𝑠𝑖𝑛 (90° − 𝜃) 𝑡𝑎𝑛 𝜃 = 𝑐𝑜𝑡 (90° − 𝜃) 𝑐𝑜𝑡 𝜃 = 𝑡𝑎𝑛 (90° − 𝜃) 𝑠𝑒𝑐 𝜃 = 𝑐𝑠𝑐 (90° − 𝜃) 𝑐𝑠𝑐 𝜃 = 𝑠𝑒𝑐 (90° − 𝜃) Trigonometric Functions of Any Angle ● If 𝑟 = √𝑥 2 + 𝑦 2 is the distance from (0,0) to (x,y), the six trig functions of 𝜃are 𝑠𝑖𝑛 𝜃 = 𝑦 𝑟 𝑐𝑠𝑐 𝜃 = 𝑐𝑜𝑠 𝜃 = 𝑥 𝑟 𝑠𝑒𝑐 𝜃 = 𝑡𝑎𝑛 𝜃 = 𝑦 ,𝑥 ≠ 0 𝑥 𝑟 ,𝑦 ≠ 0 𝑦 𝑟 .𝑥 ≠ 0 𝑥 𝑥 𝑐𝑜𝑡 𝜃 = , 𝑦 ≠ 0 𝑦 ● Reference Angle: positive acute angle between the terminal side and x-axis ● Find Reference Angles for Angles Greater Than 360°( 2𝜋) or Less Than -360° (-2𝜋) ○ Find a positive angle𝛼 less than 360° that is coterminal with the given angle. ○ Draw𝛼 in standard position. ○ Use the drawing to find the reference angle for the given angle. Graphs of Trig Functions ● The graph of 𝑦 = 𝐴 𝑠𝑖𝑛(𝐵𝑥 + 𝐶)can be obtained using amplitude |𝐴|, period 2𝜋 𝑏 , and 𝐶 phase shift 𝐵. ● The graph of 𝑦 = 𝐴 𝑐𝑜𝑠(𝐵𝑥 − 𝐶) can be obtained using amplitude |𝐴|, period 2𝜋 𝑏 𝐶 phase shift . 𝐵 ● 𝑦 = 𝐴 𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷 𝑎𝑛𝑑 𝑦 = 𝐴 𝑐𝑜𝑠(𝐵𝑥 − 𝐶) + 𝐷 , the constant D can cause vertical shifts. If 𝐷 > 0, you shift upward. If 𝐷 < 0, you shift downward. Inverse Trig Functions , and 𝜋 ● Inverse Sine Function: inverse of the restricted sine function 𝑦 = 𝑠𝑖𝑛 𝑥, − 2 ≤ 𝑥 ≤ 𝜋 2 which means 𝑦 = 𝑠𝑖𝑛−1 𝑥 𝑚𝑒𝑎𝑛𝑠 𝑠𝑖𝑛 𝑦 = 𝑥 ● Finding Exact Values of 𝑠𝑖𝑛−1 𝑥 ○ 1) Let 𝜃 = 𝑠𝑖𝑛−1 𝑥 𝜋 ○ 2) Rewrite 𝜃 = 𝑠𝑖𝑛−1 𝑥 as 𝑠𝑖𝑛𝜃 = 𝑥, 𝑤ℎ𝑒𝑟𝑒 − 2 ≤ 𝜃 ≤ 𝜋 2 ○ 3) Use the table to find the exact values that satisfies 𝑠𝑖𝑛𝜃 = 𝑥 ● Inverse Cosine Function: inverse of the restricted cosine function 𝑦 = 𝑐𝑜𝑠 𝑥, 0 ≤ 𝑥 ≤ 𝜋which means 𝑦 = 𝑐𝑜𝑠 −1 𝑥 𝑚𝑒𝑎𝑛𝑠 𝑐𝑜𝑠 𝑦 = 𝑥 ● Finding Exact Values of 𝑐𝑜𝑠 −1 𝑥 ○ 1) Let 𝜃 = 𝑐𝑜𝑠−1 𝑥 ○ 2) Rewrite 𝜃 = 𝑐𝑜𝑠−1 𝑥as 𝑐𝑜𝑠𝜃 = 𝑥, where 0 ≤ 𝜃 ≤ 𝜋 ○ 3) Use the table to find the exact values ● Inverse Tangent Function: inverse of the restricted tangent function 𝑦 = 𝑐𝑜𝑠 𝑥, 0 ≤ 𝑥 ≤ 𝜋 𝜋which means 𝑦 = 𝑡𝑎𝑛 𝑥, − 2 ≤ 𝑥 ≤ 𝜋 2 ● Finding Exact Values of 𝑡𝑎𝑛−1 𝑥 ○ 1) Let 𝜃 = 𝑡𝑎𝑛−1 𝑥 𝜋 ○ 2) Rewrite 𝜃 = 𝑡𝑎𝑛−1 𝑥 𝑎𝑠 𝑡𝑎𝑛 𝜃 = 𝑥 , 𝑤ℎ𝑒𝑟𝑒 − 2 < 𝜃 < ○ 3) Use the table to find exact values 𝜋 2 ● Graphs of Three Basic Inverse Trig Functions ❖ Analytic Trigonometry Verifying Trig Identities ● Fundamental Trig Identities ○ Reciprocal Identities 𝑠𝑖𝑛 𝑥 = 1 𝑐𝑠𝑐 𝑥 𝑐𝑜𝑠 𝑥 = 1 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 = 1 𝑐𝑜𝑡 𝑥 𝑐𝑠𝑐 𝑥 = 1 𝑠𝑖𝑛 𝑥 𝑠𝑒𝑐 𝑥 = 1 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑡 𝑥 = 1 𝑡𝑎𝑛 𝑥 ○ Quotient Identities 𝑡𝑎𝑛 𝑥 = 𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑡 𝑥 = 𝑐𝑜𝑠 𝑥 𝑠𝑖𝑛 𝑥 ○ Pythagorean Identities 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 = 1 ○ Even-Odd Functions 1 + 𝑡𝑎𝑛2 𝑥 = 𝑠𝑒𝑐 2 𝑥 1 + 𝑐𝑜𝑡 2 𝑥 = 𝑐𝑠𝑐 2 𝑥 𝑠𝑖𝑛 (−𝑥) = −𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 (−𝑥) = 𝑐𝑜𝑠 𝑥 𝑐𝑠𝑐 (−𝑥) = −𝑐𝑠𝑐 𝑥 𝑠𝑒𝑐 (−𝑥) = 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 (−𝑥) = −𝑡𝑎𝑛 𝑥 𝑐𝑜𝑡 (−x) = -cot x ● Principal Trig Identities 𝑠𝑖𝑛 (𝛼 + 𝛽) = 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 + 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽 𝑠𝑖𝑛 (𝛼 − 𝛽) = 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 − 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 (𝛼 + 𝛽) = 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 (𝛼 − 𝛽) = 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠 𝛽 + 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽 𝑡𝑎𝑛 (𝛼 + 𝛽) = 𝑡𝑎𝑛 𝛼 + 𝑡𝑎𝑛 𝛽 1 − 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛𝛽 𝑡𝑎𝑛 (𝛼 − 𝛽) = 𝑡𝑎𝑛 𝛼 − 𝑡𝑎𝑛 𝛽 1 + 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛𝛽 ○ Product To Sum Formula 1 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽 = [𝑐𝑜𝑠 (𝛼 − 𝛽) − 𝑐𝑜𝑠(𝛼 + 𝛽)] 2 1 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠 𝛽 = 2 [𝑐𝑜𝑠 (𝛼 − 𝛽) + 𝑐𝑜𝑠(𝛼 + 𝛽)] 1 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 = [𝑠𝑖𝑛 (𝛼 + 𝛽) − 𝑠𝑖𝑛(𝛼 − 𝛽)] 2 1 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽 = [𝑠𝑖𝑛 (𝛼 + 𝛽) − 𝑠𝑖𝑛(𝛼 − 𝛽)] 2 → 𝑆𝑢𝑚 𝑡𝑜 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎𝑠 ❖ Conic Sections ● The Ellipse: is the set of all points P in a plane the sum of whose distances from two fixed points is constant ○ Foci: the two fixed points ○ Center: midpoint of the segment connecting the foci ● Standard Form Of Ellipse ○ 𝑥2 𝑦2 + 2 2 𝑎 𝑏 = 1 𝑜𝑟 𝑥2 𝑏 2 2 + 𝑦𝑎2 = 1 ● Standard Form of Equations for Ellipses ○ ● The Hyperbola: a set of points in a plane the difference of whose distances from two fixed points, called foci is constant ○ Vertices: line through the foci that intersects at two points ○ Standard Forms of the Equations of a Hyperbola ■ 𝑥2 𝑦2 − 2 2 𝑎 𝑏 = 1 𝑜𝑟 𝑥2 𝑏 2 2 − 𝑦𝑎2 = 1 ○ Standard Forms of Hyperbolas Centered at (h.k) ■ ● The Parabola: set of all points in a plane that are equidistant from a fixed line. ○ Directrix: a fixed point ○ Focus: not on the line ○ Latus Rectum: line segment that passes through its focus, parallel to the directrix, the endpoints are located on the parabola ■ Latus rectum is |4𝑝| ● Standard Form of the Equation of a Parabola ○ 𝑦 2 = 4𝑝𝑥 𝑜𝑟 𝑥 2 = 4𝑝𝑦 ❖ Matrices and Determinants ● Augmented Matrices: has a vertical bar separating the columns of the matrix into 2 groups ● Row-Echelon Form: matrix with 1s down the main diagonal and 0s below the 1s ● Gaussian Elimination: process used to solve linear systems using matrix row operations ● Gauss-Jordan Elimination: reduced row-echelon form, the process for a matrix with 1s down the main diagonal and 0s in every position above and below each 1 is found ● Matrix Addition and Subtraction: matrices of the same order are added or subtracted by adding or subtracting. ○ Properties of Matrix Addition ■ 1) A+B = B+A ■ 2) ( A+B) + C = A + (B+C) ■ 3) A+0 = 0+A =A ■ 4) A + (-A) = (-A) + A = 0 ● Scalar Multiplication: product of a real number and a matrix ○ Properties of Scalar Multiplication ■ (cd)A = c(dA) ■ 1A=A ■ c(A+B) = cA+cB ■ (c+d)A = cA+dA ○ Properties of Matrix Multiplication ■ (AB)C= A(BC) ■ A(B+C)= AB+ AC ■ (A+B) C = AC + BC ■ c(AB) = (cA) B ● Finding Multiplicative Inverses for Invertible Matrices ● Finding the Determinant of a 2x2 Matrix ● Cramer’s Rule: method of using determinants to solve the linear equation ❖ Additional Topics in Trigonometry ● Law of Sines ○ 𝑎 𝑠𝑖𝑛 𝐴 = 𝑏 𝑠𝑖𝑛 𝐵 = 𝑐 𝑠𝑖𝑛 𝐶 ○ Ambiguous Case: given information may result in one triangle, two triangles, or no triangle at all ● Law of Cosine ○ 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠 𝐴 ○ 𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 𝑐𝑜𝑠 𝐵 ○ 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 𝑐𝑜𝑠 𝐶 ● Polar Coordinates: (𝑟, 𝜃) ● Rectangular Coordinates: (x,y) ● Multiple Representations of Points in the Coordinate System ○ If 𝑛is any integer, the point (𝑟, 𝜃)can be represented as (𝑟, 𝜃) = (𝑟, 𝜃 + 2𝑛𝜋) 𝑜𝑟 (𝑟, 𝜃) = (−𝑟, 𝜃 + 𝜋 + 2𝑛𝜋) ● Relations between Polar and Rectangular Coordinates ○ 𝑥 = 𝑟 𝑐𝑜𝑠 𝜃 ○ 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃 ○ 𝑥2 + 𝑦2 = 𝑟2 𝑦 ○ 𝑡𝑎𝑛𝜃 = 𝑥 ● Converting a Point from Rect. to Polar Coordinates (𝑟 > 0 𝑎𝑛𝑑 0 ≤ 𝜃 ≤ 2𝜋 ○ 1) Plot the point (x,y) ○ 2) Find r by computing the distance from the origin to (x,y): 𝑟 = √𝑥 2 + 𝑦 2 𝑦 ○ 3) Find 𝜃using tan 𝜃 = 𝑥with the terminal side passing through (x,y) ● Absolute Value of a Complex Number ○ Absolute value 𝑎 + 𝑏𝑖 is |𝑧| = |𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏 2 ● Polar Form of a Complex Number ○ The complex number 𝑧 = 𝑎 + 𝑏𝑖is written in polar form as 𝑧 = 𝑟(𝑐𝑜𝑠 𝜃 + 𝑏 𝑖 𝑠𝑖𝑛 𝜃), where 𝑎 = 𝑟 𝑐𝑜𝑠 𝜃, 𝑏 = 𝑟 𝑠𝑖𝑛 𝜃, 𝑟 = √𝑎2 + 𝑏 2 ,and 𝑡𝑎𝑛 𝜃 = . 𝑎 ■ Modulus: value of r ■ Argument: value of 𝜃 ● Product of Two Complex Numbers in Polar Forms ○ Let 𝑧1 = 𝑐𝑜𝑠𝜃1 + 𝑖 𝑠𝑖𝑛𝜃1 ) 𝑎𝑛𝑑 𝑧2 = 𝑐𝑜𝑠𝜃2 + 𝑖 𝑠𝑖𝑛𝜃2 ), the product would be 𝑧1 𝑧2 = 𝑟1 𝑟2 [𝑐𝑜𝑠(𝜃1 + 𝜃2 ) + 𝑖 𝑠𝑖𝑛(𝜃1 + 𝜃2 )]. ■ To multiply 2 complex #s, multiply the moduli and add arguments. ● Quotient of Two Complex Numbers in Polar Form ○ Let 𝑧1 = 𝑟1 (𝑐𝑜𝑠𝜃1 + 𝑖 𝑠𝑖𝑛𝜃1 ) 𝑎𝑛𝑑 𝑧2 = 𝑟2 (𝑐𝑜𝑠𝜃2 + 𝑖 𝑠𝑖𝑛𝜃2 ), the quotient would 𝑧 be 𝑧1. 2 ● DeMoivre’s Theorem: formula for the nth power and multiplying the argument by n ○ 𝐿𝑒𝑡 𝑧 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃) be a complex number in polar form. If 𝑛is a positive integer, then 𝑧to the nth power, 𝑧 𝑛 is 𝑧 𝑛 = [𝑟(𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃)]𝑛 = 𝑟 𝑛 (𝑐𝑜𝑠 𝑛𝜃 + 𝑖 𝑠𝑖𝑛 𝑛𝜃). ● Using DeMoivre’s Theorem for Finding Complex Roots ● Vectors: quantities that involve both a magnitude and a direction (usually denoted with v) ● Scalars: quantities that involve magnitude, but no direction ● i and j Unit Vectors ○ i - direction is along the positive x-axis ○ j - direction is along the positive y-axis ● Adding and Subtracting Vectors in Terms of i and j ○ If 𝑎1 𝑖 + 𝑏1 𝑗 𝑎𝑛𝑑 𝑤 = 𝑎2 𝑖 + 𝑏2 𝑗, 𝑡ℎ𝑒𝑛 𝑣 + 𝑤 = (𝑎1 + 𝑎2 )𝑖 + (𝑏1 + 𝑏2 )𝑗 𝑜𝑟 𝑣 − 𝑤 = (𝑎1 − 𝑎2 )𝑖 + (𝑏1 − 𝑏2 )𝑗 ● Scalar (k) Multiplication with a Vector in Terms of i and j ○ 𝑘𝑣 = (𝑘𝑎)𝑖 + (𝑘𝑏) 𝑗 ● Properties of Vector Addition and Scalar Multiplication ○ Vector Addition Properties ■ 𝑢+𝑣 =𝑣+𝑢 ■ (𝑢 + 𝑣) + 𝑤 = 𝑢 + (𝑣 + 𝑤) ■ 𝑢+0 =0+𝑢 = 𝑢 ■ 𝑢 + (−𝑢) = (−𝑢) + 𝑢 = 0 ○ Scalar Multiplication Properties ■ (𝑐𝑑)𝑢 = 𝑐(𝑑𝑢) ■ 𝑐(𝑢 + 𝑣) = 𝑐𝑢 + 𝑐𝑣 ■ (𝑐 + 𝑑)𝑢 = 𝑐𝑢 + 𝑑𝑢 ■ 1𝑢 = 𝑢 ■ 0𝑢 = 𝑢 ■ ||𝑐𝑣|| = |𝑐|||𝑣|| ● The Dot Product: is defined as 𝑣 ∙ 𝑤 = 𝑎1 𝑎2 + 𝑏1 𝑏2 ● Properties of the Dot Product ○ If 𝑢, 𝑣, 𝑎𝑛𝑑 𝑤are vectors, and c is a scalar then, ■ 𝑢∙𝑣 =𝑣∙𝑢 ■ 𝑢 ∙ (𝑣 + 𝑤) = 𝑢 ∙ 𝑣 + 𝑢 ∙ 𝑤 ■ 0∙𝑣 =0 ■ 𝑣 ∙ 𝑣 = ||𝑣|| 2 ■ (𝑐𝑢) ∙ 𝑣 = 𝑐(𝑢 ∙ 𝑣) = 𝑢 ∙ (𝑐𝑣) ● Alt. Formula for The Dot Product ○ 𝑣 ∙ 𝑤 = ||𝑣||||𝑤||𝑐𝑜𝑠 𝜃 ● Vector Projection of v Onto w ○ 𝑝𝑟𝑜𝑗𝑤 𝑣 = 𝑣∙𝑤 𝑤 ||𝑤|| 2 ***NOTE: We do not claim ownership of any images used in this study guide. Some definitions, examples, and pictures were used from the following sources: ● Blitzer Precalculus (Third Edition) ● Onlinemathlearning.com ● Basicmathematics.com ● ● ● ● ● Tes Tech Lumen Learning Technology UK Michael Van Biezen Mathwords