A guide to the Math Placement Test. Department of Mathematical Sciences George Mason University

A guide to the Math Placement Test Department of Mathematical Sciences George Mason University Revised November 0

Contents Introduction... Courses Requiring the Placement Test... Policies... Test Format... Learning Resources... 4 BasicAlgebra... Fractions... 6 Decimal Notation... 7 Percents... 8 Ratios.... 0 Algebraic Epressions... Linear Equations in One Variable... Linear Equations in Two Variables... Graph/Slope of a line... Algebra I... 4 Linear Inequalities in Variable... 4 Absolute Value... 4 Systems of Linear Equations... Linear Inequalities in Variables... 6 Multiplying Algebraic Epressions... 6 Factoring Algebraic Epressions... 7 Rational Epressions... 7 More on Eponents... 8 Radicals... 9 Quadratic Equations... 9 Functions Notation... 0 Equations with Rational Epressions... Algebra II... More on Functions... Graphs of Functions... 4 Transcendentals... Eponential Functions... Logarithmic Functions... 6 Angle Measure... 6 Right Triangle and Circular Functions Definitions... 7 Graphs of Trigonometric Functions... 8 Inverse Trigonometric Functions... 9 Trigonometric Formulas and Identities... 0 Dear Student, The Math Placement Test is quite likely the most important eam that you will take as you start your studies at George Mason University. Most majors require a mathematics course. Depending on which mathematics course you intend to take, the placement test will measure your preparedness for that course. Failure to earn the appropriate test score for an intended course will preclude you from registering for that course. Appropriate courses that will prepare you for your intended course will be suggested, so please make all efforts to take this test early and avoid needless frustrations later. This booklet has been prepared by Ellen O Brien, Director of the Mathematics Learning Center, to give you detailed information concerning the test. The Learning Center supports your learning of mathematics in an ancillary way to the regular classroom lectures. It provides, for eample, self paced or instructor lead tutorials for remediation in mathematics. It offers free, limited tutoring for all undergraduates enrolled in mathematics courses. These are staffed by our upper division math undergraduates and graduate students. The booklet also contains sample problems with solutions and a quick review of basic mathematical concepts to help you prepare for the Placement Test. Of course, the best way to prepare is to review the mathematics that you have learned until now. Please take this part of your preparation and the eam itself seriously. I invite you to visit our math website to view services, courses and programs that the department offers. Finally, I wish you a successful and rewarding academic career at George Mason University. Regards, David Walnut Professor and Chair Department of Mathematical Sciences Answers...

Courses that Require the Placement Test The Math Placement Test is given by the Department of Mathematical Sciences to determine the readiness for the following courses: Math 04 or Math 0 Pre-Calculus Math 0 Introductory Probability Math 08 Introductory Calculus with Business Applications Math Analytic Geometry and Calculus I Math Calculus Algebra/Trigonometry A Math Discrete Mathematics CS Computer Science I Students should talk to an academic advisor to determine which Math course(s) they are required to take. All students admitted to the university are advised to take the Math Placement Test during the orientation process. Test Format The Math Placement Test is a computer-based test in a multiple choice format. You may need to do some calculations on paper in order to determine the best answer. There is a time limit for completing each section of the test. The test is divided into four parts: Basic Algebra Algebra I Algebra II Transcendentals Students who are attempting to place into Math 0 must take the Basic Algebra section. All other students must start with the Algebra I section. The sections of the test are sequential. For eample, students must pass Algebra I in order to proceed to the Algebra II section. If you are unsure of which math course you are required to take, please consult your academic advisor. Policies For current policies and required minimum scores please visit the webpage: http://math.gmu.edu/placement_test.php

Learning Resources The Math Learning Center offers two programs designed to prepare students for their required math courses. The Basic Math Program refreshes skills needed for success in Math 0. The Algebra Program prepares students for Math 0, Math 08 or Math. Both are self-paced programs offered in an online format. http://math.gmu.edu/math-learning-center.php The center also manages the Undergraduate Mathematics Tutoring Center which specializes in general help for freshman/sophomore math classes. It is staffed by upper division mathematics majors and graduate students. http://math.gmu.edu/tutor-center.php Both the Mathematics Tutoring Center and the Math Learning Center are located in Johnson Center Room 44. Hours are posted on the webpage above. We are closed on holidays. More information about the Math Placement Test is available at http://math.gmu.edu/placement_test.php Basic Algebra The Basic Algebra section of the test measures your ability to perform basic arithmetic and Algebraic operations and to solve problems that involve these operations. The following eercises and eamples provide a sample of the type of material that is tested in the Basic Algebra section. Only those students attempting to place into Math 0 will be given questions from the Basic Algebra section of the test. Basic Operations Eample: John s car insurance premium is $.00. He has the option of making four equal payments but will be charged an additional $ processing fee. If he chooses the four-payment option, what is the amount of each payment? $.00 The processing fee is added to the premium +.00 This amount is then divided by 4 Each payment will be $8.00 $ 40.00 8 4 40 Eercise : A shipment of 44 CD s is to be packed into cartons containing 4 CD s each. How many cartons are necessary to ship these CD s? Eercise : A loan of $0,00.00 is to be paid off in 60 equal monthly payments How much is each payment (ecluding interest)? Eercise : Susan s Cell phone provider charges her $.00 per month for the first 00 minutes of calls and $0.70 for each minute over 00. If she uses 40 minutes of calls in a given month, how much will she be charged? 4

Fractions A fraction or rational number is of the form q p where p and q are integers, and q 0 p is called the numerator and q is called the denominator. The fraction is said to be reduced to lowest terms if p and q have no factors (other than ) in common. The same number can be epressed as a fraction in many ways. These fractions are called equivalent fractions:,,, 0, 4 0 40 48 Only the first fraction is in reduced form. To change from a mied number to an improper fraction, multiply the whole number by the denominator of the fraction and add the result to the numerator of the fraction. This gives the numerator of the improper fraction. Eample: 8 4 8 ( ( ) + ) ( 4 ( 8) + ) a c ac The product of two fractions is. That is, multiply numerators to b d bd get the numerator of the product and multiply denominators to get the denominator of the product. Eercise 4: Perform the indicated operation and give the result in reduced form. a. b. 6 c. d. 6 8 4 8 The sum of two fractions with a common denominator is a b a b + + c c c If fractions do not have the same denominator then they must be changed to an equivalent form. Eample: Find the sum + 8 Change each fraction to an equivalent form 4 8 8 4 8 8 4 Then add the equivalent fractions to get 8 4 Eercise : form a. 47 + + or 4 4 4 Perform the indicated operation and give the result in reduced 7 6 + b. 4 8 7 + c. 0 4 d. 8 6 4 7 Eamples: 4 9 8 99 6 6 The quotient of two fraction a c is equal to b d a d or b c ad cb Decimal Notation Rational numbers or fractions can also be epressed in decimal notation. 7 can be written as 7. 00. is equivalent to or 000 8 Eample: 49 ounces of a solution is to be poured into test tubes with capacity ounces. How many test tubes will be filled? 7 7 49 49 49 4 4 test tubes will be filled. When adding or subtracting numbers in decimal notation, make sure to align the decimal point. Eample: Find the sum.+. 78 6 7

+..78 6.88 When multiplying two numbers in decimal notation, first multiply the numbers disregarding the decimal point(s). Net put the decimal point in the product. The number of places to the right of the decimal point in the product should be the SUM of the places to the right of the decimal point in each of the factors. Eample: Find the product 6.. 6.. 6 700 The product should have three places to the right of the decimal point. The result will be. Eample: Carlos is getting paid $0.80 per hour and time and a half for hours over 40 per week. If he works 4. hours this week, how much will he earn? Carlos is paid $ 0.80. $6. 0 per hour of overtime He will earn: ( 40 $0.80) + (. $6.0) 488. 70 Percents The symbol % is used to indicate the fraction or the ratio: to 00 00 % is equivalent to or. 00 00 00% is equivalent to or. 00 00 Eercise 7: Change the percent to a decimal a. % b. % c. 0.% Eample: A class contains female and 8 male students. What percent of the class is female? There are a total of 0 students in the class. out of 0 or are female 0 60 60% of the class is female 0 00 Eercise 8: Find the following a. 0% of 7 b. 4% of 96 c. 0% of 00 Eample: Suppose 60% of a class is female. Of these females, 0% are freshman. What percent of the class is female upper class students? If 0% of the females are freshman, then 80% of females are upper class students. 80% of 60%. 80.60. 48 so 48% of the class are female upper class students Eample: Suppose we know that 40% of a class or 6 students are male. How many students are in the class? We know 40% of (the number) is 6 That is. 40 ( the number) 6 So 6 ( the number ) 40 students.40 Eample: Before graduating, Joe worked in a convenience store and earned $.8/hr. He started a new job that is paying him $8.9/hr. What is the percent increase in his hourly rate? actual increase % increase so 8.9.8 or % original.8 Eercise 6: Change the decimal to percent a.. b..08 c.. 8 9

Ratio The ratio of two numbers a and b is written a to b, a:b or b a We can think of this as a fraction in order to simplify it. The ratio 0: is equivalent to :... 6 The ratio. to is equivalent to 6 to 0 since 0 To show two ratios are equal use the following: A C if and only if AD BC B D Eample: Show that. : is equivalent to the ratio :. if and only if. or Eample: How much water should be mied with 7 cups of salt to make a % saline solution? 7 We have the proportion part 00 total Multiply to get 700 8 So we need 8 7 cups of water. Eponents When working with algebraic epressions involving eponents, the following properties of eponents are used:.. a b a+ b a a a 4. ( y ) y a a b. b b a ab. ( ) y a y 6. 0 for any 0 a a 0 Eercise 9: Simplify each of the following epressions using the rules of eponents 4 a. ( ) 0 b. 4a c. 6 y ab y 7 0 6 y z y z Eercise 0: Find the product ( ) Algebraic Epressions ( ) Most of Algebra involves manipulating epressions that contain variables. Eample: Evaluate the epression using the indicated value of the variable ( k )( k + ) for k ( ( ) ) (( ) + ) ( 6 )( + ) ( 7)( ) - substitute - in the epression for k Eercise : Evaluate the epression using the indicated value of the variable + a. 6 + for b. for 4 4 + ( ) Eercise : Combine the terms below and simplify the result. a. 4( + ) + 8 b. 6 a b + 4( a + b) c. 4 + 0 Linear Equations in One Variable A linear equation in one variable can have a unique solution, no solution or infinitely many solutions. When it has no solution, the equation is called inconsistent.

Eample: Solve the following equations by isolating the variable. 4 + + 4 4 + + 4 Add 4 to both sides 6 6 + + Add to both sides 6 6 6 6 Multiply both sides by 6 6 6 6 Eample: Solve the equation 6 4( 4 ) The equation has a unique solution 6 6 Distribute the 4 6 6 6 6 Add -6 to both sides 0 The equation is inconsistent. Eample: Solve the equation 0 + ( ) ( 6 0) + + 0 + 0 + + Clear the parentheses by distributing Combine like terms 0 0 Since the last statement is always true, every real number will be a solution. Linear Equations in Two Variables A solution for an equation in two variables is a replacement for each of the variables so that when substituted, the resulting statement is true. We generally write the solution as an ordered pair. For eample, (,) is a solution of the equation + y. When the variable is replaced by in the equation and the variable y is replaced by, the resulting statement is true. The linear equation in two variables has an infinite number of solutions. The graph of these solutions is a line in the -y plane. Eample: The formula in two variables gives the conversion between!! Fahrenheit and Celsius temperatures. ( C to y F) 9 y + Find the temperature in degrees Celsius that corresponds to! F 9 + Substitute the value for y 9 80 Add - to both sides The Graph of a Line 00 Multiply both sides by 9 A solution of the equation is ( 00,) We can graph the solutions of any linear equation on a set of aes called the Euclidean Plane. In the case of the linear equation, the solutions form a line.\we can measure how much and in which direction the line tips using the slope. The slope-intercept form of the equation is : y m + b If the equation is in this form, the line has slope m and y-intercept (0, b) The y-intercept is the point where the line intersects the y-ais. Eample: In the graph below, find the and y intercepts and the slope. Eample: Find the slope and y-intercept of the line with equation: 0 y Write the equation in slope-intercept form: y The slope is and the y-intercept 0, Parallel lines have identical slopes. Lines are perpendicular if their slopes have the relationship m m intercept: (-, 0) y intercept (0, 6) Slope is /

Eample: Find the slope of a line that is perpendicular to the line 0 y The line 0 y has slope. Any line perpendicular to it will have slope m -. Algebra I The Algebra I section of the test measures your ability to perform basic algebraic operations and to solve problems that involve algebraic concepts. Only those students attempting to place into Math 08, Math, Math, Math, CS will be given questions from the Algebra I section. In addition to all material in the Basic Algebra, the following eercises and eamples provide a sample of the type of material that is tested in the Algebra I section. Linear Inequalities in One Variable Unlike linear equations, linear inequalities have an infinite number of solutions. We can use a number line or interval notation to indicate the solutions. Any real number can be added to or subtracted from both sides of the inequality. We can multiply or divide both sides by any nonzero real number but, if we multiply or divide by a negative number, the inequality sign must be reversed. Eample: Solve the inequality 9 + Subtract from both sides: 9 Multiply both sides by -/9: The solution is the interval:, 9 9 Absolute Value Equations and Inequalities The absolute value of a number is defined to be its distance from 0 on the number line. for 0 for < 0 Eample: Solve the equation for 7 7 or 7 0 or 0 4 Eample: Solve the inequality + + or + 0 or Distance Formula The distance between two points (, y ) and ( ) the formula: ( ) ( ) d + y y The formula uses the Pythagorean Theorem, y is given by Eercise : Find the distance between the points (,) and (,8) Systems of Linear Equations We sometimes group equations together and investigate whether they have a common solution. This is called a system of equations. The system of linear equations has a solution if there is an ordered pair (, y) that satisfies each equation in the system. A system of linear equations can have eactly one solution, no solution or infinitely many solutions. There are several methods available to solve systems of linear equations. The net eample illustrates the Elimination Method. + y 48 Eample: Solve the system 9 8y 4 + y 48 multiply by -: 9 6y 44 9 8y 4 9 8y 4 Add: 4 y 68 Solve for y: y The value is substituted for y in either of the original equations. + Solve the resulting equation for ( ) 48 + 4 48 4 8 Eercise : Solve the systems y a. + y 0 The solution to the system is (8, ) y b. 6 y 0

Linear Inequalities in Two Variables Linear inequalities in two variables can be solved by graphing. The solution is an infinite set of ordered pairs in the plane. We can indicate the solution by shading the region that contains these ordered pairs. Eample: Solve the inequality y Graph the line that results from replacing the inequality sign with an equal sign. Here, graph the line y This line partitions the plane into three sets: The solutions of y > y < y In the inequality, test an ordered pair from the plane that does not lie on the line. Using (0,0), we get: (0) (0) which is a true statement. Shade the half-plane of points that satisfy the inequality. See the graph below. To multiply binomial factors, we apply the Distributive Law twice and combine like terms. Eample: Find the product ( + 7)( ) ( + 7)( ) ( ) + 7( ) + 7 + 7 ( ) + 4 7 + 7 The result of this multiplication is called a quadratic epression. We undo the multiplication in a process called factoring. Factoring Algebraic Epressions The net two eamples show the factoring process. In the first, we factor out the highest common factor among the terms. In the second, we write the quadratic epression as a product of binomial factors. In a similar way we can solve a system of inequalities. The graph below shows $ y & the solutions to the system % y 0 & ' 0 Multiplying Algebraic Epressions Eample: Factor the epression 6 + 4 6 + 4 ( 6 + 4) The terms have the facto in common 8 ( + ) They also have a factor o 8 in common Eample: Factor the epression t + 8t 40 + 8t 40 t + 0 t Special Factorings: a b ( a + b)( a b) a b ( a b)( a + ab + b ) a + b ( a + b)( a ab + b ) t ( )( ) We use the Distributive Law to multiply algebraic epressions. Eample: Find the product 6 ( 0) 6 ( 0) 6 ( ) + 6( 0) 60 6 Rational Epressions Rational epressions are fractions where the numerator and/or denominator involve variables. We can reduce, add, subtract, multiply and divide these rational epressions using the rules of fractions. You may want to review the rules of fractions in the Basic Algebra section on page 6. 7

7 Eample: Reduce the rational epression 0 In order to reduce the epression we must factor both the numerator and the denominator, then cancel any factors common to both. 7 7( ) 0 ( + )( ) 7( ) Factor the numerator ( + )( ) 7 Cancel the common factor + Eample: Find the sum 4 + + 9 ( ) 4 + + 9 4 + ( + )( + ) ( + )( ) ( + )( ) 4 + ( + 4 + ) 9 ( + )( ) Eample: Simplify the epression a + b Using the rules of dividing fractions we get: a + b ( a + b) More Eponents Use a common denominator The definition of eponents is etended to include all rational numbers: a a Eercise : If a b find the value of - Eercise 4: Find the value of b 7 9 a 8 Radicals When simplifying algebraic epressions involving radicals, the following properties of radicals are used: n y. n n. y n n where y 0 n y y m n m. ( ) n Eample: Find the product ( 6 + ) ( ) ( 6 ) ( ) 6 + + 6 Eercise : Find the sum: 8 + 7 Eercise 6: Simplify each of the following epressions 6 a. 8a b b. 7 8u v 7uv Eercise 7: Rationalize the denominator to simplify: Quadratic Equations + 8 y c. An equation of the form a + b + c 0 is called a quadratic equation. The equation will have two, one or no real number solutions. There are several methods available to find the solutions of a quadratic equation. The following eamples illustrate these methods. Eample: Solve the equation 0 0 0 Isolate the term involving the radical 0 Take the square root of both side of the equation 9

Eample: Solve the equation + 4 + 4 + 4 + 4 + 4 0 bring all terms to one side ( + 7 )( ) 0 factor the left side + 7 0 or 0 set each factor equal to zero 7 or the equation has solutions Eample: Solve the equation 9 + 6 + 0 In the quadratic formula: a 9, b 6 and c b ± b 4ac a we use 6 ± 6 4 9 6 since b 4ac 0 9 8 we get only one solution to the equation. The epression b 4ac is called the discriminant. It tells us how many solutions a quadratic equation will have. If b 4ac > 0 the equation has real number solution If b 4ac 0 the equation has real number solution If b 4ac < 0 the equation has no real number solution Eercise 8: Solve the equations a. 4 49 0 b. + 9 0 c. + 4 6 0 Function Notation Functions can be defined using a list of ordered pairs, a graph, a table, or an equation. The following equation defines a function: f ( ). f. Each value of is paired with the value ( ) f ( ) 8 so the pair (,8) f ( 0) 0 99 so the pair (,99) f ( 0) ( 0) 99 so the pair ( 0,99) is in the function f. 0 is in the function f. is in the function f. 0 + ( ) + Eample: For the function f ( ) find f ( ) and f ( b +) f ( ) f ( b + ) Comple Rational Epression ( b + ) b + b ( b + ) + b + 4 A comple rational epression is one in which a rational epression appears in the numerator and/or denominator. Simplifying the comple rational epression is equivalent to finding the quotient of the epression in the numerator with the epression in the denominator. Eample: Simplify the comple rational epression y y Combine the terms in the denominator first. y y y Net, find the quotient of the numerator with the denominator y y y y y y y Equations with Rational Epressions Given an equation involving rational epressions we can find an equivalent equation with no rational epressions. To find the equivalent equation we multiply the entire equation by the least common denominator. After solving the equivalent equation, we must check the solution in the original to see if it makes any denominator equal to zero. If so, we discard that solution. Eample: Solve the equation + The LCD of the rational epressions is 0 0 0 + 0 Multiply equation by 0

0 + Solve the resulting equation 4 Eample: Solve the equation The LCD of the rational epressions is 4 Multiply equation by 4 Solve the resulting equation + 4 0 Use the quadratic formula ± 4 ( 4) ± 49 4 and Eercise 9: Solve the equation Algebra II + 6 + + The Algebra II section of the test measures your ability to solve problems that involve more advanced Algebra than the previous section. Only those students attempting to place into Math and CS will be given questions from the Algebra II section. The following eercises and eamples provide a sample of the type of material that is tested in the Algebra II section. More on Functions A function is a set of ordered pairs where each first component is paired with eactly one second component. The set of first components is called the domain and the set of second components is called the range. Unless otherwise stated, the domain of the function f ( ), is all real numbers. Any real number a can be paired with another real number using the equation ( ) a, a. f. The pair can be written ( ) Suppose we have the function g( t). We say ( t) t g is undefined at t 0 since the epression is undefined. On the other hand, g ( t) is defined at any 0 real number other than 0. The domain of ( t) Suppose we have the function f ( ) + g is all real numbers ecept t 0.. In order for the epression + to represent a real number, the epression + must not be negative. For this function the domain is all real numbers. The average rate of change for a function f() is the change in f() over the change in. Δf ( ) average rate of change Δ Alternatively, it is the slope of the secant line passing through the two points (, f( )) and (, f( )) f ; Find each of the following: + 6 a. the domain of f ( ) b. the range of f() c. the average rate of change from - to Eercise : Let ( ) Eercise : Let g( ) ; Find each of the following: a. the domain of g ( ) b. the range of g() c. the average rate of change from - to We can compose two functions by applying one right after the other. Eample: For the functions f and g defined in the eercises and, find g 6 g f a. ( ( ) ( g( 6 ) f b. ( ( ) f f f ( 6) 0 g ( f ( ) g g ( ) + 6 or /

Eample: The function values of f and g are given in the table below. Use them g 0 g f to find a. f ( ( )) b. ( ( )) a. f ( g( 0) ) ( ) b. g ( f ( ) ) ( ) f() g() 0-0 - 0 f g Some functions have an inverse. The inverse of a function will interchange the real numbers in the ordered pair. If ( a, b) is in the function f then ( b, a) is in f, the inverse of f. Keep in mind, that not all functions have inverses. The functions f ( ) + and f ( ) are inverses. Notice f contains the pair (,4) and f - contains the pair (4, ). This is true for all ordered pairs of the function f. Graphs of Functions We can graph the set of ordered pairs of a function on a set of aes just as we do with linear equations in variables. The graph of a quadratic function f ( ) a + b + c is a parabola. The verte of the parabola is located at b b, f ( ) a a This point on the graph is either a minimum or a maimum for the function. The leading coefficient a, determines which. For a<0, the function takes on a b maimum at and the range of the function is b y y f. a a For a>0, the function takes on a minimum at b and the range of the a function will be b y y f. a 4 Eample: In the graph of f ( ) 4 + 7 below, find the and y intercepts and the verte. To graph the function y c f ( a) + b we shift the graph of f() right a units (for a>0), left a units for a<0) up b units for b>0, down b units for b<0 vertically stretch the graph of f() by a factor of c, c> vertically shrink the graph of f() by a factor of c, for 0<c< y Eample: The graph above can be viewed as a shift of ( units right and units up). The function can be written in the form: f ( ) ( ) + Transcendentals y intercept: (0, 7) intercept: none verte: (, ) note: (, ) is the minimum of f() The test measures your ability to work with eponential, logarithmic and trigonometric functions of the right triangle and the unit circle. The graphs of these functions and some common trigonometric formulas are given in the net few pages. Eponential Functions An eponential function is one in which the variable occurs in the eponent. The base determines whether the function is increasing or decreasing. f ( ) a for a > f ( ) a for a < Domain: all real numbers Domain: all real numbers Range: y>0 Range: y>0

Logarithmic Functions Right Triangle Definitions A logarithmic function is the inverse of an eponential function. This is the natural log function, (the inverse of ye ) f() ln Domain: >0 Range: all real numbers opp sin θ hyp adj cos θ hyp tan θ opp adj hyp csc θ opp hyp sec θ adj cot θ adj opp Eample: Evaluate: a. log 8 b. log c. log 0. 00 a. log 8 since 8 b. log 0 since 0 c. log 0.00 since 0. 00 Eample: Solve the equation for : e + 0 Isolate by taking the log of both sides. Here use ln the natural log. lne + ln0 + ln0 + ln0. 6974 log 4 ( + ) Eample: Solve the equation for : 6 Use the inverse function to isolate log ( ) 6 4 4 + 4 6 + 4 408 Angle Measure! An angle can be measured in degrees or in radians. A full revolution is 60 or π radians. We use this information to convert from one measure to the other. The following proportion shows the relationship between degrees and y radians. 60 y π Eample: Find the degree measure of an angle whose radian measure is π 60 ( π / ) π π 60( π /) so 0 6 Definitions of Circular Functions Special Angles Degree Radian sin θ cos θ tan θ csc θ sec θ cot θ 0 0 0 0 -- -- 0 π/6 / 4 π/4 60 π/ / / / / / / / / 90 π/ 0 -- -- 0 Sum and Difference Formulas ( u ± v) sin( u) cos( v) cos( u) sin( v) ( u ± v) cos( u) cos( v) sin( u) sin( v) sin ± cos ± / 7

Graphs of the Trigonometric Functions f()sin() period: π Inverse Trigonometric Functions The inverse sine function, written sin ( ) or arcsin( ), has domain [-,] and range π π,. It is defined to be the number in the interval π π, whose sin is. In a similar way, cos ( ) or arccos() is defined to be the number 0 whose cosine is. The inverse tangent function, written in the interval [,π ] tan ( ) whose tangent is. or arctan( ), is defined to be the number is the interval π π, f()cos() period: π Eercise : Find each of the following: a. π arctan b. arccos(cos ) c. arcsin( 0) f()tan() period: π Eercise : Find the period of sin b. cos c. tan 4 a. ( ) Eercise : Which has the shortest period? b. y cos c. y cos( ) d. y cos a. y cos 8 9

Double Angle Formulas ( u) sin( u) cos( u) ( u ) cos ( u) sin ( u) sin ( u) cos ( u) sin cos Law of Sines sin A sin B sin C a b c Law of Cosines c a + b Pythagorean Identity sin + cos ab cosc Eercise 4: In the triangle below, find the length of the side marked with. Eercise : In the triangle below, find the length of the side marked with. Answers to Eercises Basic Algebra. 6. 67. $6.00 4. (a) 7 (b) (c) (d) 0 6 0. 9 (a) 7 (b) (c) (d) 8 0 4 6. (a) % (b) 8 % (c) % 7. (a).0 (b).0 (c).00 8. (a). (b) 7.84 (c) 60 9. (a) 8 9 4 60 y z (b) a b 0.. (a) (b) /80 9 (c) y. (a) 4 4 (b) 0 a + 0b (c) Algebra I. 8 or. (a) (0, -) (b) inconsistent system 7. 4. or 7. 7 8 6. (a) a b b 7. 4 + 6 4 0 (b) u v (c) y 7 7 8. (a), 9., 6 (b), (c) 0, + 0 Eercise 6: If sin A and tan A, find cos A Algebra II. (a) all real numbers ecept (b) all real numbers ecept y0 (c) -/. (a) all real numbers (b) all real numbers y>0 (c) or 4 < ( ) 4 0

Transcendentals. (a) (b) 8π (c) π. c. (a) 4. π! 0 (b)! sin 70 π (c) 0. 6.