General Algebra Study Guide PDF

Created by Sion and Alexis for Use by Simple Studies General Algebra (Summary of Algebra 1 & 2) Study Guide Topic 1: Fundamentals of Algebra Basics of Solving Equations: ○ Vocabulary Numerical Expression: an expression is a phrase used in math that has...

Created by Sion and Alexis for Use by Simple Studies General Algebra (Summary of Algebra 1 & 2) Study Guide Topic 1: Fundamentals of Algebra Basics of Solving Equations: ○ Vocabulary Numerical Expression: an expression is a phrase used in math that has numbers and operations, but no an equal sign Equation: a mathematical sentence that uses an equal sign; separates two equivalent expressions Solution: a value that takes place of a variable value, making an equation true Operations: Addition, Subtraction, Multiplication, Division, etc. Properties of Equality: properties proving that if you do an operation on both sides of an equation, the statement will remain true ○ How to Solve Two-Step Equations 1. Isolate the variable (move everything else to the other side of the equal sign) 2. Use the opposite of PEMDAS to solve (SADMEP) - think of this like unraveling the equation a. Everything you do to one side you must also do to the other 3. Simplify all the way until you have a numerical solution to your variable Example: 6x - 4 = 12 6x - 4 + 4 = 12 + 4 6x = 16 6x/6 = 16/6 x = 8/3 Set of Numbers: ○ Vocabulary Roster Notation: all numbers in a set listed between braces, {}  Created by Sion and Alexis for Use by Simple Studies Interval Notation: () are used to exclude the endpoint of an interval while [] are used to include the endpoint of an interval ex. (2, 6) all real numbers between but not including 2 and 6 [1, 3] all real numbers between and including 1 and 3 Square Roots: ○ Vocabulary Radical Symbol: a symbol that shows that the number on the right squared is equal to the number inside the radical √36 = 6 ○ Properties of Square Roots Product Property of Square Roots: √𝑎𝑏 = √𝑎 ⋅ √𝑏 𝑎 𝑎 Quotient Property of Square Roots: √√𝑏 = √𝑏 Simplifying Algebraic Expressions: ○ Content To evaluate expressions with given values of the variables, substitute the numbers into the correct variable and use PEMDAS to solve To simplify algebraic expressions, first identify like terms by the variable and the exponents, then combine like terms. Exponents: ○ Vocabulary Scientific Notation: a way of writing large numbers by using powers of 10 ex. 3.274 х 10⁶ = 3,274,000 ○ Properties of Square Roots Zero Exponent Property: a⁰ = 1 −𝑛 1 1 Negative Exponent Property: 𝑎 = (𝑎)𝑛 = 𝑛 𝑎 𝑎 −𝑛 𝑏 𝑛 (𝑏 ) = (𝑎) Product of Powers Property: am・an = am +n Quotient of Powers Property: am/an = am-n Power of a Power Property: (am)n = am・n Power of a Product Property: (ab)m = ambm  Created by Sion and Alexis for Use by Simple Studies Power of a Quotient Property: (a/b)m = am/bm Transformations: ○ Vocabulary Transformation: changing a geometric figure by altering its size, shape or position in order to create a new figure Translation: a type of transformation where a figure slides across a graph, keeping the size and shape of the figure Reflection: a type of transformation where a figure is reflected over a line of reflection, typically over the x or y-axis Stretch: pulling points away from the x or y-axis Compression: pushing points towards the x or y-axis ○ Translations Horizontal Translation: each point moves right or left (x, y) → (x + h, y) Vertical Translation: each point moves up or down (x, y) → (x, y + k) ○ Reflections Reflection across x-axis: y-coordinate changes (x, y) → (x, -y) Reflection across y-axis: x-coordinate changes (x, y) → (-x, y) ○ Stretches and Compressions Horizontal Stretch: (x, y) → (bx, y) |b|>1 Vertical Stretch: (x, y) → (x, ay) | a | >1 Horizontal Compression: (x, y) → (bx, y), 0< |b| Greater than < Less than ≥ Greater than or equal to ≤ Less than or equal to ≠ Not equal ○ Graphing Inequalities  Created by Sion and Alexis for Use by Simple Studies (picture from https://www.algebra-class.com/solving-inequalities.html) ○ Graphing Systems of Linear Inequalities Graph the lines just like you would for linear equations. Then shade one side of the line, depending on your inequality’s sign (if the equation is greater than y - shade above, if it is less than - shade below) Open Circle: Dashed/Dotted Line Closed Circle: Regular Line After shading all of the inequalities of the system, the area that is shaded by all of them is the solution Curve Fitting with Linear Models: ○ Vocabulary Regression: statistical study of the relationship between variables to find trends Correlation: a measure of the strength of the relationship between a pair of variables as well as what direction the relationship leans toward  Created by Sion and Alexis for Use by Simple Studies Line of Best Fit: a line that best represents the data according to each point Correlation Coefficient: represented by r-measures how well two variables fit the correlation (-1 ≤ r ≥ 1) ○ Correlation Coefficient r = 1, straight line, positive slope r = 0, no correlation r = -1, straight line, negative slope Absolute-Value Equations and Inequalities: ○ Vocabulary Disjunction: compound statement that joins two statements with the word OR (∪) Conjunction: compound statement that joins two statements with the word AND (∩) Absolute-value function: a function that contains an absolute value sign (parent absolute value function has a v-shape with its point at the origin) Linear Programming: ○ Vocabulary Linear Programming: a way of finding the biggest or smallest value of a function that matches a given set of conditions called constraints Constraint: one inequality in a problem; limits Feasible Region: a graphed solution of a constraint Objective Function: finds the best combination in a constraint whether it is a minimum or maximum, both or neither values ○ Content Find the maximum value by identifying the vertices and putting each into the equation.  Created by Sion and Alexis for Use by Simple Studies Topic 3: Exponential Functions and Relationships The Basics of Radicals ○ Vocabulary Radical Expression: and expression that has a radical, one that uses the square root operation (√) Radicand: the number inside of the radical sign, usually referred to as n Index: the number sitting on top of the tail of the racial sign, usually referred to as a  Created by Sion and Alexis for Use by Simple Studies Rational Exponent: the number inside of the radical sign, usually referred to as n ○ Radical Properties Product Property of Radicals: √ab = √a * √b Quotient Property of Radicals: √a/b = √a/√b Exponential Functions, Growth, and Decay ○ Vocabulary Asymptote: a line of a graphed function that could never be approached as the line continues Exponential Function: a function where the outputs values are related to each other by a constant ratio; modeled with the parent function: y=abx Exponential Growth: a function where a is greater than zero and b is greater than one; the values grow exponentially and rapidly Exponential Decay: when b is a number between 0 and 1; the y- values decrease exponentially as the x-values get larger ○ Exponential Regression A tool usually found on graphing calculators that helps you find the exponential function and whether it is growth or decay Takes points and graphs them to form a line and equation for the regression Topic 4: Polynomial Functions and Operations Polynomials ○ Vocabulary Polynomial Equations: an equation consisting of many terms with multiple different powers using the same variables Tri(nominal): consisting of three terms Poly(nominal): consisting of four or more terms Adding and Subtracting Polynomials ○ Remember! Rewrite the equation, pairing like terms together and turning double subtraction into addition  Created by Sion and Alexis for Use by Simple Studies Multiplying Polynomials ○ Tips and Tricks Making tables helps when multiplying a lot of terms Utilize the Distributive Property ○ FOIL Method Multiply the First terms Multiply the Outer terms Multiply the Inner terms Multiply the Last terms Dividing Polynomials: ○ Vocabulary Synthetic division: a way of dividing polynomials by linear binomial using coefficients instead of using long division ○ Content  Created by Sion and Alexis for Use by Simple Studies (picture from https://www.onlinemathlearning.com/ dividing-polynomials-synthetic-division-2.html) Factoring Polynomials: ○ Factor Theorem You can use this theorem to determine whether a linear binomial is a factors Use synthetic substitution to solve. The binomial is a factor of P(x) only if P(x) equals 0 ○ Factoring by Grouping Another method to solve polynomials 1) Group terms 2) Factor out common variables or numbers from each group 3) Factor out common binomial 4) Factor the difference of squares ○ Factoring the Sum and the Difference of Two Cubes  Created by Sion and Alexis for Use by Simple Studies Sum of two cubes: a3 + b3 = (a + b)(a2 - ab + b2) Difference of two cubes: a3 - b3 = (a - b)(a2 + ab + b2) Finding Real Roots of Polynomial Equations ○ Vocabulary Multiplicity: the number of times a root is repeated on a single point ○ Multiplicity When a root has a multiplicity that is bigger than 1, the graph bends at the point of the root After you have factored a polynomial equation, you can determine what multiplicity of a root has by seeing how many of the binomial appears ex. (x-3)(x-3)(x+4)➝root 3 has a multiplicity of 2 and -4 has a multiplicity of 1 ○ Rational Root Theorem Every Rational Root of a Polynomial could be written in the form p/q, where p represents the factor of the polynomial and q is the factor of the leading coefficient of the polynomial ○ Irrational Root Theorem When a + b√c is a root of a polynomial equation, then a - b√c is also a root of the polynomial P(x) = 0. Fundamental Theorem of Algebra: ○ The Fundamental Theorem of Algebra Every polynomial with a degree of 1 or larger has at least one zero, where a zero could be a complex number ○ Complex Conjugate Root Theorem If a + bi is a root of a polynomial equation, so is a - bi. Graphs of Polynomial Functions: ○ Vocabulary End Behavior: the behavior of a polynomial function as x approaches positive or negative infinity Turning Point: where a graph turns from decreasing to increasing or increasing to decreasing  Created by Sion and Alexis for Use by Simple Studies Local Maximum: the highest point given a range Local Minimum: the lowest point given a range Transforming Polynomial Functions: ○ Content (picture from https://slideplayer.com/slide/7334686/) Topic 5: Quadratic Functions and Inequalities Graphing Quadratic Functions ○ Vocabulary Quadratic Function: a function that can be represented in this form: f(x) = ax2 + bx + c Vertex Form: f(x) = a(x - h)2 + k , where (h, k) is the vertex  Created by Sion and Alexis for Use by Simple Studies Quadratic Formula: Parabola: the curve of a graphed quadratic function Vertex: the point on the parabola where the graph changes direction Axis of Symmetry: a line that divides the parabola in half, it passes through the vertex Found using the following equation: x = - b/2a Minimum: the value of the vertex when it is the lowest point of the graph (when a > 0) Maximum: the value of the vertex when it is the highest point of the graph (when a < 0) ○ Maximums and Minimums (picture from https://www.mymatheducator.com/ gr-11-lesson-3-maximum-or-minimum-of-a-quadratic-function.html) ○ Solving for Zeros Zero: an x value that accounts for a parabola’s x-intercept value (x, 0) Solve by factoring from the ax2 + bx + c to two factors Utilize the Zero Product Property 1. Set each factor’s f(x) equal to zero 2. Solve each of the equations for x to find the zeros.  Created by Sion and Alexis for Use by Simple Studies (picture from https://slideplayer.com/slide/10907022/) Solving Quadratic Inequalities: ○ Vocabulary Quadratic Inequality in two variables: inequalities that require two variables; they can be written in any of these forms: y≤ ax2 + bx + c y≥ ax2 + bx + c yax2 + bx + c ○ Graphing Quadratic Inequalities 1) Graph the Quadratic inequality just like you would in a quadratic equation 2) Use solid line for y≤ and y≥ and a dashed line for y< and y> 3) Shade above the line for y≥ and y> and below for y≤ and y< Operations with Complex Numbers ○ Vocabulary Complex Plane: a graph where the x-axis represents real numbers and the y-axis represents imaginary numbers Absolute Value of a Complex Number: a + bi equals the distance from the point (a,b) in the complex plane  Created by Sion and Alexis for Use by Simple Studies Topic 6: Logarithmic Functions Logarithmic Functions: ○ Vocabulary Logarithm: the quantity representing the power to which the base has to be raised in order to meet a certain number bx = a → logba = x  Created by Sion and Alexis for Use by Simple Studies Common Logarithm: if there is no base of a logarithm, assume it’s 10 Logarithmic Function: inverse of an exponential function ○ Content Logarithm is basically the opposite of exponents: 26 = 64 → log264 = 6 When solving for log, ask yourself: what exponent of the base will meet the power Special Properties of Logarithms: Logarithmic Form Exponential Form Example Logarithm of Base b log2020 = 1 1 logbb = 1 b =b 201 = 20 Logarithm of 1 log101 = 0 logb1 = 0 b0 = 1 100 = 1 Exponential vs. Logarithmic Function (picture from https://miprofe.com/en/tag/funcion-exponencial/) Properties of Logarithms: ○ Product Property of Logarithms The sum of the logarithms of its factors is equal to the logarithm of a product log45000 = log4(50 ・100) = log450 + log4100  Created by Sion and Alexis for Use by Simple Studies logbmn = logbm + logbn ○ Quotient Property of Logarithms The logarithm of a quotient= the difference between the logarithm of the dividend and the logarithm of the divisor 12 log4( 2 ) = log4(12) - log4(2) 𝑚 logb( 𝑛 ) = logbm - logbn ○ Power Property of Logarithms The logarithm of a power equals the logarithm of the base log104 log (10)(10)(10)(10) log10 + log10 + log10 + log10 4log10 ○ Inverse Properties of Logarithms and Exponents logbbx = x ex. log10108 = 8 blogbx = x ex. 10log⏨3 = 3 ○ Change of Base Formula logbx = logax/logab ex. log5100 = log10100/log105 The Natural Base, e: ○ Vocabulary Natural Logarithm: a logarithm with a base e (abbreviated “In”) Natural Logarithmic Function: Inverse of the natural exponential function f(x) = ex natural log function→ (f(x) = In x) ○ Content Use properties of logarithms and apply them to solve natural logarithms Transformation of Logarithmic Functions ○ Content Transformation f(x) Notation Examples Vertical Translation f(x) + k y = log x + 4 4 units up y = log x - 3 3 units  Created by Sion and Alexis for Use by Simple Studies down Horizontal Translation f (x-h) y = log(x - 7) 7 units right y = log(x + 2) 2 units left Vertical Stretch af(x) y = 2 log x stretch by Or Compression 2 y = ¼ log x compression by ¼ Horizontal Stretch or f(1/b x) y = log (⅙x) stretch by Compression 6 y = log (2x) compression by ½ Topic 7: Probability, Sequences, and Statistics Patterns and Sequences: ○ Vocabulary Sequence: a group of numbers that are listed in a specific order, due to some pattern that comes with each term Term: a number in a sequence Position Number: each number in a sequence has a position number based on what number of the sequence they are (ex. 7, 9, 11, 13...is a sequence, 11 is the 3rd number so 11’s position number is 3)  Created by Sion and Alexis for Use by Simple Studies Explicit Rule: a rule that can be used to find the nth term of a sequence that does not have depending factors on previous terms, you do not need to find all of the previous terms to find some nth term (ex. f(n) = n2 + 2) Recursive Rule: a rule used to find the nth term of a sequence that has factors depending on previous terms, you cannot skip around and must find every previous term before the nth term you are looking for, these are more complex than explicit rules (ex. f(n) = f(n - 1) + 3) Statistical Models: ○ Vocabulary Quantitative Data: data that can be expressed using numerical values Categorical Data: data that cannot be expressed using numerical values Frequency: how often something appears in a set of data, is usually organized in a table Two-Way Frequency: frequencies with two variables ○ Relative Frequency You can convert data to its relative frequency to know what part or percentage belongs to a certain group Relate frequency can be found by dividing the amount in the group by the total amount of frequencies Joint Relative Frequency Marginal Relative Conditional Relative Frequency Frequency A percentage that tells you A percentage that tells you A percentage that tells you what portion of the total data what portion of the data fits what portion of the one fits into both of the two into one of the categories that category’s data fits into categories that are being are being looked at another category looked at Measures of Center ○ Vocabulary  Created by Sion and Alexis for Use by Simple Studies Mean: the sum of the values in a data set divided by the number of values in the set Range: the difference between the biggest and smallest value points Median: the value within the middle of a data set ○ Box Plots (picture from https://www.simplypsychology.org/boxplots.html) Whisker: lines extending from the box of a box plot, connecting it to the minimum and maximum values Minimum: the smallest value of data displayed on a box plot and in a set of data Lower/First Quartile: the median between the minimum and the actual median Interquartile Range (IQR): the difference between the third and first quartile data points Upper/Third Quartile: the median between the actual median and the maximum Maximum: the largest value of data displayed on a box plot and in a set of data Outlier: a data point that is exceedingly far from the other data points, usually leads to the skewing of data  Created by Sion and Alexis for Use by Simple Studies Standard Deviation ○ Vocabulary Mean Standard Deviation: the square root of the average distance away from the mean by each data point Normal Distribution ○ Vocabulary Normal Distribution: data that when displayed, is symmetric, looks like a bell-shaped curve with tails on either side, a normal curve (picture from https://www.investopedia.com/terms/b/bell-curve.asp) Scatter Plots and Trend Lines ○ Vocabulary Scatter Plot: a visual showcasing two-variable data Two-Variable Data: a collection of paired values such as ordered pairs Line of Best Fit: a line that is drawn through a scatter plot, showing the correlation of the two-variable data Residual: a vertical measure of the space between a data point and the line of best fit Linear Regression: the most widely used method to find the least-squares line Least-Squares Line: line that represents the sum of the squared residuals as the smallest it can be  Created by Sion and Alexis for Use by Simple Studies Permutations and Combinations: ○ Vocabulary Fundamental Counting Principle: a method to figure out the number of outcomes of an event; you label the number of ways to choose an item as m1・m2・.... mn when n represents the item Permutation: listing all the ways a group of items could be placed Factorial: Multiplying every natural number under a number until one including itself (5! = 5・4・3・2・1 = 120) Combination: grouping items in no certain order ○ Content (picture from: https://www.dummies.com/education/math/algebra/algebraic- permutations-and-combinations/) Theoretical and Experimental Probability: ○ Vocabulary Probability: how likely something is to occur Theoretical Probability: ratio of the number of favorable outcomes to the total number of outcomes for equally likely outcomes Complement: the probability of event A’s outcomes except itself (P(not A) = 1 - P(A) Geometric Probability: probability solved by a ratio of lengths, areas, or volumes Experimental Probability: the quotient of the number of times the event occurs and the number of trials Independent and Dependent Events:  Created by Sion and Alexis for Use by Simple Studies ○ Vocabulary Independent Events: an event by itself that does not affect another event’s probability Dependent Events: when the occurence of one event affects the probability of another event Conditional Probability: states how to solve the measure of the probability of an event given that another event happened ○ Probability of Independent Events When A and B are both independent events, then P(A and B) = P(A)・P(B) ○ Probability of Dependent Events When both A and B are dependent events, P(A and B) = P(A)・P(B | A), where P(B | A) is the probability of B, given that A has occurred Compound Events: ○ Vocabulary Simple Event: an event with one outcome Compound Event: several simple events Mutually Exclusive Events: events that cannot happen at the same time in an experiment Inclusive Events: events that can happen simultaneously  Created by Sion and Alexis for Use by Simple Studies Mutually Exclusive Events (picture from https://slideplayer.com/slide/13416593/) ○ Inclusive Events  Created by Sion and Alexis for Use by Simple Studies (picture from https://slideplayer.com/slide/5825670/) Binomial Distributions: ○ Vocabulary Binomial Theorem: an easy way to expand any binomial using Pascal’s triangle Binomial Experiment: experiment that has independent trials with successful or failing outcomes Binomial Probability: P(r) = nCrprqn-r Series and Summation Notation: ○ Vocabulary Partial Sum: sum of terms in part of the sequence Summation Notation: a sign that helps with effectively adding sequences ○ Content  Created by Sion and Alexis for Use by Simple Studies (picture from https://www.slideserve.com/matt/summation-notation) Arithmetic Sequences and Series: ○ Vocabulary Arithmetic Sequence: numbers in which the difference between those terms are constant Arithmetic Series: sum of terms in an arithmetic sequence ○ Content (picture from https://www.bisd303.org/cms/lib3/WA01001636/Centricity/ Domain/574/Algebra%202/Alg%202B%20Notes/Alg2%20Notes%209.3.pdf) Mathematical Induction and Infinite Geometric Series: ○ Vocabulary Infinite Geometric Series: series that have infinitely many terms Converge: when a series is |r| < 1 and the partial sum approaches a fixed number Limit: the number that the partial sums approach as n increases  Created by Sion and Alexis for Use by Simple Studies Diverge: when a series is |r| ≥ 1 and the partial sum does not approach a fixed number Mathematical Induction: a type of mathematical proof to prove formula you used for sums ○ Content Sum of an Infinite Geometric Series: S = a1/1-r’ when r is the common ratio and |r| < 1 is where a1 is the first term Topic 8: Rational and Radical Functions Rational Functions: ○ Vocabulary  Created by Sion and Alexis for Use by Simple Studies Rational Function: a function that could be written as a ratio of two polynomials within which its graph is a hyperbola ○ Content (picture from http://p2cdn3static.sharpschool.com/UserFiles/ Servers/Server_2879574/File/teacher- documents/Math/Wesson/Algebra%203/Notes/Chapter%205/Notes%205- 4.pdf) (picture from http://p2cdn3static.sharpschool.com/UserFiles/ Servers/Server_2879574/File/teacher- documents/Math/Wesson/Algebra%203/Notes/Chapter%205/Notes%205- 4.pdf) (picture from http://p2cdn3static.sharpschool.com/UserFiles/  Created by Sion and Alexis for Use by Simple Studies Servers/Server_2879574/File/teacher- documents/Math/Wesson/Algebra%203/Notes/Chapter%205/Notes%205- 4.pdf) Radical Expressions and Rational Exponents: ○ Vocabulary Index: the n of a radical expression n√a Rational Exponent: an exponent that is a fraction a1/n = n√a Radical Functions: ○ Vocabulary Radical Function: a function that includes a radical expression Square-root Function: a radical function with a square root ○ Content (picture from https://slideplayer.com/slide/7527692/) Radical Equations: ○ Vocabulary Radical Equation: an equation that contains a variable inside a radical ○ Solving Radical Equations  Created by Sion and Alexis for Use by Simple Studies 1) Isolate the radical/variable 2) Undo the radical by raising both sides of the equation to the power equal to the index of the radical 3) Simplify and solve Topic 9: Matrices Matrices and Data: ○ Vocabulary Matrix: a rectangular array of numbers inside a bracket Dimensions: “m by n”; m rows and n columns Scalar: Multiplying a matrix by a number ○ Adding and Subtracting Matrices You can only add or subtract matrices only if they have the same dimensions Add or subtract the number in the matrix corresponding to the spot its in ［2 3］+［4 1］= ［6 4］ ○ Scalar You multiply every number in the matrix by the scalar to get the scalar product Multiplying Matrices: ○ Vocabulary Matrix Product: multiplying two or more matrices ○ Multiplying Matrices You can only multiply matrices when the number of columns of the first and the number of rows in the second are equal  Created by Sion and Alexis for Use by Simple Studies (picture from http://web.wapak.org/hs/2Math/Mrs.% 20Rogers/Advanced%20Trig/1-4-11%20Matrices%20Day%202.pdf) Determinants and Cramer’s Rule: ○ Vocabulary Determinant: difference of the products of the diagonals of a 2 by 2 matrix ○ Cramer’s Rule for Two Equations (picture from http://www.statisticslectures.com/topics/cramersrule/)  Created by Sion and Alexis for Use by Simple Studies Topic 10: Conic Sections Conic Sections: ○ Vocabulary Conic Sections: a section formed where a double right cone and a plane intersect ○ Content You can get the following shapes in a conic section: circles, ellipses, hyperbolas, and parabolas Circles: ○ Vocabulary Tangent: a line that touches only a single point of a circle ○ Equation of a Circle (x - h)2 + (y - k)2 = r2 (h, k) is the center of the circle and r is the radius Hyperbolas: ○ Vocabulary Hyperbola: a symmetrical curve formed by the intersection of two equal cones ○ Content How the standard form of the hyperbola is written determines whether the hyperbola’s transverse axis is horizontal or vertical 𝑥2 𝑦2 Horizontal Equation: 𝑎2 − 2 = 1 𝑏  Created by Sion and Alexis for Use by Simple Studies 𝑦2 𝑥2 Vertical Equation: 𝑎2 − 2 = 1 𝑏 Identifying Conic Sections: ○ Content (picture from https://slideplayer.com/slide/756447/)  Created by Sion and Alexis for Use by Simple Studies Topic 11: Trigonometric Functions, Graphs and Identities Right-Angle Trigonometry ○ Vocabulary Trigonometric Function: a function that relates to a ratio of sides of a right triangle Sine: opposite/hypotenuse Cosine: adjacent/hypotenuse Tangent: opposite/adjacent Cosecant: 1/sinθ = hypotenuse/opposite Secant: 1/cosθ = hypotenuse/adjacent Cotangent: 1/tanθ = adjacent/opposite The Unit Circle: ○ Vocabulary Radian: unit of angle measure of an arc when arc length is equal to the radius Unit Circle: a circle that has a radius of 1 (picture from https://www.mathsisfun.com/geometry/unit-circle.html) ○ Converting Angle Measures Degrees to radians: multiply degrees by (π radians/180°)  Created by Sion and Alexis for Use by Simple Studies Radians to Degrees: multiply radians by (180°/π radians ○ Using Reference Angles to Evaluate Trigonometric Functions 1) Find the reference angle θ (smallest positive angle made with the x-axis) 2) Get the sine, cosine, and tangent by applying the reference angle 3) Change the signs according to the original quadrant of the angle if needed The Law of Sines: ○ Area of a Triangle You can use these formulas to solve for the area of a triangle when you don’t know the height of the triangle For triangle ABC, where abc are sides opposite to the angles ABC: Area = ½bc sinA Area = ½ bc sin B Area = ½ab sin C ○ Law of Sines For triangle ABC, with sides abc, the Law of Sines says that: sinA = sinB = sinC a b c The Law of Cosines: ○ Content For triangle ABC, where corresponding sides of the angles are abc, Law of Cosines states: a2 = b2 + c2 - 2bccosA b2 = a2 + c2 - 2accosB c2 = a2 + b2 - 2abcosC ○ Heron’s Formula  Created by Sion and Alexis for Use by Simple Studies (picture from https://www.onlinemathlearning.com/herons-formula.html) Graphs of Trigonometric Functions: ○ Vocabulary Periodic Functions: functions that repeat cycles in regular intervals Amplitude: half of the difference of the maximum and minimum values of sine and cosine functions Frequency: a way to measure sound by the number of cycles in a given unit of time Fundamental Trigonometric Identities: ○ Reciprocal Identities cscθ = 1/sinθ secθ = 1/cosθ cotθ = 1/tanθ ○ Tangent and Cotangent Ratio Identities Tanθ = sinθ/cosθ cotθ = cosθ/sinθ ○ Pythagorean Identities cos2θ + sin2θ = 1 1 + tan2θ = sec2θ cot2 + 1 = csc2θ  Created by Sion and Alexis for Use by Simple Studies ○ Negative-Angle Identities sin(-θ) = -sinθ cos(-θ) = cosθ tan(-θ) = -tanθ Sum and Difference Identities: ○ Vocabulary Rotation Matrix: rotating a point through an angle θ using matrices ○ Sum Identities sin(A + B) = sinAcosB + cosAsinB cos(A + B) = cosAcosB - sinAsinB tan(A + B) = tanA +tanB/1 - tanAtanB ○ Difference Identities sin(A - B) = sinAcosB - cosAsinB cos(A - B) = cosAcosB + sinAsinB tan(A - B) = tanA -tanB/1 + tanAtanB ○ Rotation Matrix (picture from https://www.slideserve.com/madonna-holman/14-4)