JEFFERSON MATH PROJECT REGENTS BY PERFORMANCE INDICATOR: TOPIC

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1 JEFFERSON MATH PROJECT REGENTS BY PERFORMANCE INDICATOR: TOPIC NY Algebra /Trigonometry Regents Exam Questions from Fall, 009 to June, 0 Sorted by PI: Topic Dear Sir I have to acknolege the reciept of your favor of May. in which you mention that you have finished the 6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a further pursuit of that branch of science would be useful to you. there are some propositions in the latter books of Euclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted with them. trigonometry, so far as this, is most valuable to every man, there is scarcely a day in which he will not resort to it for some of the purposes of common life. the science of calculation also is indispensible as far as the extraction of the square & cube roots; Algebra as far as the quadratic equation & the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be indulged in by one who is to have a profession to follow for his subsistence. in this light I view the conic sections, curves of the higher orders, perhaps even spherical trigonometry, Algebraical operations beyond the d dimension, and fluxions. Letter from Thomas Jefferson to William G. Munford, Monticello, June 8, 799.

2 TABLE OF CONTENTS TOPIC PI: SUBTOPIC QUESTION NUMBER GRAPHS AND STATISTICS A.S.-: Analysis of Data... - A.S.: Central Tendency... 5 A.S.: Dispersion A.S.6-7: Regression A.S.8: Correlation Coefficient... A.S.5: Normal Distributions PROBABILITY A.S.0: Permutations A.S.: Combinations A.S.9: Differentiating Permutations and Combinations... - A.S.: Sample Space... A.S.: Geometric Probability... 5 A.S.5: Binomial Probability ABSOLUTE VALUE A.A.: Absolute Value Equations... 0 A.A.: Absolute Value Inequalities... - QUADRATICS A.A.0-: Roots of Quadratics A.A.7: Factoring Polynomials A.A.7: Factoring the Difference of Perfect Squares... 0 A.A.5: Quadratic Formula... - A.A.: Using the Discriminant A.A.: Completing the Square A.A.: Quadratic Inequalities SYSTEMS A.A.: Quadratic-Linear Systems POWERS RADICALS A.N.: Operations with Polynomials A.A.8-9: Negative and Fractional Exponents A.A.: Evaluating Exponential Expressions A.A.8: Evaluating Logarithmic Expressions A.A.5: Graphing Exponential Functions A.A.5: Graphing Logarithmic Functions A.A.9: Properties of Logarithms A.A.8: Logarithmic Equations A.A.7: Exponential Equations A.A.6: Binomial Expansions A.A.6, 50: Solving Polynomial Equations A.N., A.: Operations with Radicals A.N.5, A.5: Rationalizing Denominators A.A.: Solving Radicals A.A.0-: Exponents as Radicals A.N.6: Square Roots of Negative Numbers A.N.7: Imaginary Numbers A.N.8: Conjugates of Complex Numbers A.N.9: Multiplication and Division of Complex Numbers i-

3 RATIONALS A.A.: Solving Rationals A.A.7: Complex Fractions A.A.5: Inverse Variation FUNCTIONS A.A.0-: Functional Notation A.A.5: Families of Functions... A.A.5: Identifying the Equation of a Graph... - A.A.8, : Defining Functions A.A.9, 5: Domain and Range... - A.A.: Compositions of Functions A.A.: Inverse of Functions A.A.6: Transformations with Functions and Relations SEQUENCES AND SERIES TRIGONOMETRY A.A.9-: Sequences A.N.0, A.: Sigma Notation A.A.5: Series... 5 A.A.55: Trigonometric Ratios A.M.-: Radian Measure A.A.60: Unit Circle A.A.6, 66: Determining Trigonometric Functions A.A.6: Using Inverse Trigonometric Functions A.A.57: Reference Angles... 6 A.A.6: Arc Length... 6 A.A.58: Cofunction and Reciprocal Trigonometric Functions 6-6 A.A.67: Proving Trigonometric Identities A.A.76: Angle Sum and Difference Identities A.A.77: Double and Half Angle Identities A.A.68: Trigonometric Equations A.A.69: Properties of Trigonometric Functions A.A.65, 70-7: Graphing Trigonometric Functions A.A.6: Domain and Range... 8 A.A.7: Using Trigonometry to Find Area A.A.7: Law of Sines A.A.75: Law of Sines - The Ambiguous Case A.A.7: Law of Cosines A.A.7: Vectors... 9 CONICS A.A.7, 9: Equations of Circles ii-

4 GRAPHS AND STATISTICS A.S.-: ANALYSIS OF DATA Howard collected fish eggs from a pond behind his house so he could determine whether sunlight had an effect on how many of the eggs hatched. After he collected the eggs, he divided them into two tanks. He put both tanks outside near the pond, and he covered one of the tanks with a box to block out all sunlight. State whether Howard's investigation was an example of a controlled experiment, an observation, or a survey. Justify your response. Which task is not a component of an observational study? The researcher decides who will make up the sample. The researcher analyzes the data received from the sample. The researcher gathers data from the sample, using surveys or taking measurements. The researcher divides the sample into two groups, with one group acting as a control group. A doctor wants to test the effectiveness of a new drug on her patients. She separates her sample of patients into two groups and administers the drug to only one of these groups. She then compares the results. Which type of study best describes this situation? census survey observation controlled experiment A survey completed at a large university asked,000 students to estimate the average number of hours they spend studying each week. Every tenth student entering the library was surveyed. The data showed that the mean number of hours that students spend studying was 5.7 per week. Which characteristic of the survey could create a bias in the results? the size of the sample the size of the population the method of analyzing the data the method of choosing the students who were surveyed A.S.: CENTRAL TENDENCY 5 The number of minutes students took to complete a quiz is summarized in the table below. If the mean number of minutes was 7, which equation could be used to calculate the value of x? 7 = 9 + x x 9 + 6x 7 = x 7 = 6 + x 6 + x 6 + 6x 7 = 6 + x

5 A.S.: DISPERSION 6 The table below shows the first-quarter averages for Mr. Harper s statistics class. A.S.6-7: REGRESSION 8 Samantha constructs the scatter plot below from a set of data. What is the population variance for this set of data? Based on her scatter plot, which regression model would be most appropriate? exponential linear logarithmic power 9 The table below shows the results of an experiment involving the growth of bacteria. 7 The scores of one class on the Unit mathematics test are shown in the table below. Write a power regression equation for this set of data, rounding all values to three decimal places. Using this equation, predict the bacteria s growth, to the nearest integer, after 5 minutes. Find the population standard deviation of these scores, to the nearest tenth.

6 0 The table below shows the number of new stores in a coffee shop chain that opened during the years 986 through 99. An amateur bowler calculated his bowling average for the season. If the data are normally distributed, about how many of his 50 games were within one standard deviation of the mean? 7 8 Assume that the ages of first-year college students are normally distributed with a mean of 9 years and standard deviation of year. To the nearest integer, find the percentage of first-year college students who are between the ages of 8 years and 0 years, inclusive. To the nearest integer, find the percentage of first-year college students who are 0 years old or older. Using x = to represent the year 986 and y to represent the number of new stores, write the exponential regression equation for these data. Round all values to the nearest thousandth. A.S.8: CORRELATION COEFFICIENT Which value of r represents data with a strong negative linear correlation between two variables? A.S.5: NORMAL DISTRIBUTIONS The lengths of 00 pipes have a normal distribution with a mean of 0. inches and a standard deviation of 0. inch. If one of the pipes measures exactly 0. inches, its length lies below the 6 th percentile between the 50 th and 8 th percentiles between the 6 th and 50 th percentiles above the 8 th percentile 5 In a study of 8 video game players, the researchers found that the ages of these players were normally distributed, with a mean age of 7 years and a standard deviation of years. Determine if there were 5 video game players in this study over the age of 0. Justify your answer. PROBABILITY A.S.0: PERMUTATIONS 6 Which formula can be used to determine the total number of different eight-letter arrangements that can be formed using the letters in the word DEADLINE? 8! 8!! 8!!+! 8!!! 7 Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA.

7 8 The letters of any word can be rearranged. Carol believes that the number of different 9-letter arrangements of the word TENNESSEE is greater than the number of different 7-letter arrangements of the word VERMONT. Is she correct? Justify your answer. 9 A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5? 8 50,0,50 A.S.: COMBINATIONS 0 The principal would like to assemble a committee of 8 students from the 5-member student council. How many different committees can be chosen? 0 6,5,,00 59,59,00 Ms. Bell's mathematics class consists of sophomores, 0 juniors, and 5 seniors. How many different ways can Ms. Bell create a four-member committee of juniors if each junior has an equal chance of being selected? 0,876 5,00 9,0 A.S.9: DIFFERENTIATING BETWEEN PERMUTATIONS AND COMBINATIONS Twenty different cameras will be assigned to several boxes. Three cameras will be randomly selected and assigned to box A. Which expression can be used to calculate the number of ways that three cameras can be assigned to box A? 0! 0!! C 0 P 0 Three marbles are to be drawn at random, without replacement, from a bag containing 5 red marbles, 0 blue marbles, and 5 white marbles. Which expression can be used to calculate the probability of drawing red marbles and white marble from the bag? 5 C 5 C 0 C 5 P 5 P 0 C 5 C 5 C 0 P 5 P 5 P 0 P A.S.: SAMPLE SPACE A committee of 5 members is to be randomly selected from a group of 9 teachers and 0 students. Determine how many different committees can be formed if members must be teachers and members must be students.

8 A.S.: GEOMETRIC PROBABILITY 5 A dartboard is shown in the diagram below. The two lines intersect at the center of the circle, and the central angle in sector measures π. 7 A study shows that 5% of the fish caught in a local lake had high levels of mercury. Suppose that 0 fish were caught from this lake. Find, to the nearest tenth of a percent, the probability that at least 8 of the 0 fish caught did not contain high levels of mercury. 8 The probability that the Stormville Sluggers will win a baseball game is. Determine the probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games. 9 The probability that a professional baseball player will get a hit is. Calculate the exact probability that he will get at least hits in 5 attempts. If darts thrown at this board are equally likely to land anywhere on the board, what is the probability that a dart that hits the board will land in either sector or sector? 6 A.S.5: BINOMIAL PROBABILITY 6 The members of a men s club have a choice of wearing black or red vests to their club meetings. A study done over a period of many years determined that the percentage of black vests worn is 60%. If there are 0 men at a club meeting on a given night, what is the probability, to the nearest thousandth, that at least 8 of the vests worn will be black? ABSOLUTE VALUE A.A.: ABSOLUTE VALUE EQUATIONS 0 What is the solution set of the equation a + 6 a = 0? 0 0, A.A.: ABSOLUTE VALUE INEQUALITIES Which graph represents the solution set of 6x 7 5? 5

9 Graph the inequality 6 x < 5 for x. Graph the solution on the line below. QUADRATICS A.A.0-: ROOTS OF QUADRATICS Find the sum and product of the roots of the equation 5x + x = 0. For which equation does the sum of the roots equal and the product of the roots equal? x 8x + = 0 x + 8x + = 0 x x 8 = 0 x + x = 0 5 For which equation does the sum of the roots equal and the product of the roots equal? x + x = 0 x x + = 0 x + 6x + = 0 x 6x + = 0 6 Write a quadratic equation such that the sum of its roots is 6 and the product of its roots is 7. A.A.7: FACTORING POLYNOMIALS 7 Factored completely, the expression 6x x x is equivalent to x(x + )(x ) x(x )(x + ) x(x )(x + ) x(x + )(x ) 8 Factored completely, the expression x + 0x x is equivalent to x (x + 6)(x ) (x + x)(x x) x (x )(x + ) x (x + )(x ) 9 Factor completely: 0ax ax 5a A.A.7: FACTORING THE DIFFERENCE OF PERFECT SQUARES 0 Factor the expression t 8 75t completely. A.A.5: QUADRATIC FORMULA The solutions of the equation y y = 9 are ± i ± i 5 ± 5 ± 5 The roots of the equation x + 7x = 0 are and and 7 ± 7 7 ± 7 A.A.: USING THE DISCRIMINANT The roots of the equation 9x + x = 0 are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal 6

10 The roots of the equation x 0x + 5 = 0 are imaginary real and irrational real, rational, and equal real, rational, and unequal A.A.: QUADRATIC INEQUALITIES 9 Which graph best represents the inequality y + 6 x x? 5 Use the discriminant to determine all value of k that would result in the equation x kx + = 0 having equal roots. A.A.: COMPLETING THE SQUARE 6 Brian correctly used a method of completing the square to solve the equation x + 7x = 0. Brian s first step was to rewrite the equation as x + 7x =. He then added a number to both sides of the equation. Which number did he add? If x + = 6x is solved by completing the square, an intermediate step would be (x + ) = 7 (x ) = 7 (x ) = (x 6) = 8 Solve x x + = 0 by completing the square, expressing the result in simplest radical form. 7

11 50 The solution set of the inequality x x > 0 is {x < x < 5} {x 0 < x < } {x x < or x > 5} {x x < 5 or x > } SYSTEMS A.A.: QUADRATIC-LINEAR SYSTEMS 5 Which values of x are in the solution set of the following system of equations? y = x 6 0, 0, 6, 6, y = x x 6 5 Solve the following systems of equations algebraically: 5 = y x x = 7x + y + POWERS A.N.: OPERATIONS WITH POLYNOMIALS 5 When x x is subtracted from 5 x x +, the difference is x + x 5 x x + 5 x x x x 5 Express x as a trinomial. 55 Express the product of y y and y + 5 as a trinomial. A.A.8-9: NEGATIVE AND FRACTIONAL EXPONENTS 56 The expression a b a 6 b 5 b 5 a 6 a b a b is equivalent to a b 57 If a = and b =, what is the value of the expression a b? When simplified, the expression equivalent to w 7 w w 7 w w 5 w 9 59 When x is divided by x, the quotient is x x (x ) is 8

12 x y 5 60 Simplify the expression and write the (x y 7 ) answer using only positive exponents. A.A.: EVALUATING EXPONENTIAL EXPRESSIONS 6 Matt places $,00 in an investment account earning an annual rate of 6.5%, compounded continuously. Using the formula V = Pe rt, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, that Matt will have in the account after 0 years. A.A.5: GRAPHING EXPONENTIAL FUNCTIONS 65 The graph of the equation y = x has an asymptote. On the grid below, sketch the graph of y = x and write the equation of this asymptote. 6 Evaluate e xln y when x = and y =. A.A.8: EVALUATING LOGARITHMIC EXPRESSIONS 6 The expression log 8 6 is equivalent to The expression log 5 is equivalent to 5 9

13 A.A.5: GRAPHING LOGARITHMIC FUNCTIONS 66 If a function is defined by the equation f(x) = x, which graph represents the inverse of this function? A.A.9: PROPERTIES OF LOGARITHMS 67 The expression logx (logy + logz) is equivalent to log x y z log x z y log x yz log xz y A B 68 If r =, then logr can be represented by C 6 loga + logb logc (loga + logb logc) log(a + B) C loga + logb logc A.A.8: LOGARITHMIC EQUATIONS 69 What is the value of x in the equation log 5 x =? ,0 70 What is the solution of the equation log (5x) =? Solve algebraically for x: log x + x + x x = 0

14 7 The temperature, T, of a given cup of hot chocolate after it has been cooling for t minutes can best be modeled by the function below, where T 0 is the temperature of the room and k is a constant. ln(t T 0 ) = kt +.78 A cup of hot chocolate is placed in a room that has a temperature of 68. After minutes, the temperature of the hot chocolate is 50. Compute the value of k to the nearest thousandth. [Only an algebraic solution can receive full credit.] Using this value of k, find the temperature, T, of this cup of hot chocolate if it has been sitting in this room for a total of 0 minutes. Express your answer to the nearest degree. [Only an algebraic solution can receive full credit.] A.A.7: EXPONENTIAL EQUATIONS 7 The solution set of x + x = 6 is {,} {,} {, } {, } 7 The value of x in the equation x + 5 = 8 x is What is the value of x in the equation 9 x + = 7 x +? 76 Solve algebraically for x: 6 x + = 6 x + 77 Akeem invests $5,000 in an account that pays.75% annual interest compounded continuously. Using the formula A = Pe rt, where A = the amount in the account after t years, P = principal invested, and r = the annual interest rate, how many years, to the nearest tenth, will it take for Akeem s investment to triple? A.A.6: BINOMIAL EXPANSIONS 78 What is the coefficient of the fourth term in the expansion of (a b) 9? 5, ,76 79 What is the fourth term in the expansion of (x ) 5? 70x 0x 70x,080x 80 Write the binomial expansion of (x ) 5 as a polynomial in simplest form. A.A.6, 50: SOLVING POLYNOMIAL EQUATIONS 8 Which values of x are solutions of the equation x + x x = 0? 0,, 0,, 0,, 0,, 8 Solve the equation 8x + x 8x 9 = 0 algebraically for all values of x.

15 8 The graph of y = x x + x + 6 is shown below. 8 The graph of y = f(x) is shown below. What is the product of the roots of the equation x x + x + 6 = 0? Which set lists all the real solutions of f(x) = 0? {,} {,} {,0,} {,0,} A.N., A.: OPERATIONS WITH RADICALS 85 The product of ( + 5) and ( 5) is The expression ab b a 8b + 7ab 6b is equivalent to ab 6b 6ab b 5ab + 7ab 6b 5ab b + 7ab 6b 87 Express 08x 5 y 8 6xy 5 in simplest radical form.

16 88 Express 5 x 7x in simplest radical form. A.N.5, A.5: RATIONALIZING DENOMINATORS 89 The fraction a b b ab b ab a 90 The expression x + x + (x + ) x x (x + ) x x x x + is equivalent to a b is equivalent to 9 Which expression is equivalent to ? 9 The expression is equivalent to 5 5 (5 ) (5 + ) Express with a rational denominator, in simplest radical form. A.A.: SOLVING RADICALS 9 The solution set of the equation x + = x is {} {0} {,6} {,} 95 The solution set of x + 6 = x + is {,} {,} {} { } A.A.0-: EXPONENTS AS RADICALS 5 96 The expression x x 5 x 5 x 5 x is equivalent to

17 97 The expression (x ) (x ) (x ) (x ) (x ) is equivalent to 98 The expression 6x y 7 is equivalent to x y 7 x 8 y 8 x y 7 x 8 y 8 A.N.6: SQUARE ROOTS OF NEGATIVE NUMBERS 99 In simplest form, 00 is equivalent to i 0 5i 0i i 5 A.N.7: IMAGINARY NUMBERS 00 The product of i 7 and i 5 is equivalent to i i 0 The expression i + i is equivalent to i i + i + i A.N.8: CONJUGATES OF COMPLEX NUMBERS 0 What is the conjugate of + i? + i i i + i 0 The conjugate of 7 5i is 7 5i 7 + 5i 7 5i 7 + 5i A.N.9: MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS 0 The expression ( 7i) is equivalent to 0 + 0i 0 i i 58 i RATIONALS A.A.: SOLVING RATIONALS 05 Solve for x: x x = + x 06 Solve algebraically for x: A.A.7: COMPLEX FRACTIONS x + x = x 9 07 Written in simplest form, the expression equivalent to x x x x x + x x x + is

18 08 Express in simplest form: d d + d A.A.5: INVERSE VARIATION 09 For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9. FUNCTIONS A.A.0-: FAMILIES OF FUNCTIONS 0 The equation y sinθ = may be rewritten as f(y) = sinx + f(y) = sinθ + f(x) = sinθ + f(θ) = sinθ + If fx = x, what is the value of f( 0)? x 6 A.A.5: FAMILIES OF FUNCTIONS On January, a share of a certain stock cost $80. Each month thereafter, the cost of a share of this stock decreased by one-third. If x represents the time, in months, and y represents the cost of the stock, in dollars, which graph best represents the cost of a share over the following 5 months?

19 A.A.5: IDENTIFYING THE EQUATION OF A GRAPH Four points on the graph of the function f(x) are shown below. {(0,),(,),(,),(,8)} Which equation represents f(x)? f(x) = x f(x) = x f(x) = x + f(x) = log x A.A.8, : DEFINING FUNCTIONS 5 Which graph does not represent a function? Which equation is represented by the graph below? y = 5 x y = 0.5 x y = 5 x y = 0.5 x 6

20 6 Which graph does not represent a function? 7 Which graph represents a relation that is not a function? 8 Which relation is not a function? (x ) + y = x + x + y = x + y = xy = 7

21 9 Which function is not one-to-one? {(0,),(,),(,),(,)} {(0,0),(,),(,),(,)} {(0,),(,0),(,),(,)} {(0,),(,0),(,0),(,)} 0 Which graph represents a one-to-one function? What is the range of f(x) = (x + ) + 7? y y y = 7 y 7 What are the domain and the range of the function shown in the graph below? {x x > }; {y y > } {x x }; {y y } {x x > }; {y y > } {x x }; {y y } The graph below represents the function y = f(x). A.A.9, 5: DOMAIN AND RANGE What is the domain of the function f(x) = x +? (, ) (, ) [, ) [, ) State the domain and range of this function. 8

22 A.A.: COMPOSITIONS OF FUNCTIONS 5 If f(x) = x and g(x) = x + 5, what is the value of (g f)()? If f(x) = x 5 and g(x) = 6x, then g(f(x)) is equal to 6x 0x 6x 0 6x 5 x + 6x 5 A.A.6: TRANSFORMATIONS WITH FUNCTIONS AND RELATIONS 0 The graph below shows the function f(x). Which graph represents the function f(x + )? 7 If f(x) = x 6 and g(x) = x, determine the value of (g f)( ). A.A.: INVERSE OF FUNCTIONS 8 Which two functions are inverse functions of each other? f(x) = sinx and g(x) = cos(x) f(x) = + 8x and g(x) = 8x f(x) = e x and g(x) = lnx f(x) = x and g(x) = x + 9 If f(x) = x 6, find f (x). 9

23 The minimum point on the graph of the equation y = f(x) is (, ). What is the minimum point on the graph of the equation y = f(x) + 5? (,) (, 8) (, ) ( 6, ) SEQUENCES AND SERIES A.A.9-: SEQUENCES What is the formula for the nth term of the sequence 5,8,6,...? a n = 6 n a n = 6 n a n = 5 n a n = 5 n What is a formula for the nth term of sequence B shown below? B = 0,,,6,... b n = 8 + n b n = 0 + n b n = 0() n b n = 0() n What is the common difference of the arithmetic sequence 5,8,,? Which arithmetic sequence has a common difference of? {0,n,8n,n,... } {n,n,6n,6n,... } {n +,n + 5,n + 9,n +,... } {n +,n + 6,n + 6,n + 56,... } 6 What is the common ratio of the geometric sequence whose first term is 7 and fourth term is 6? What is the fifteenth term of the sequence 5, 0,0, 0,80,...? 6,80 8,90 8,90 7,680 8 What is the fifteenth term of the geometric sequence 5, 0, 5,...? Find the first four terms of the recursive sequence defined below. a = a n = a (n ) n 0

24 A.N.0, A.: SIGMA NOTATION 0 The value of the expression ( r + r) is r = The value of the expression (n + n ) is 6 n = 0 A.A.5: SERIES 5 An auditorium has rows of seats. The first row has 8 seats, and each succeeding row has two more seats than the previous row. How many seats are in the auditorium? TRIGONOMETRY A.A.55: TRIGONOMETRIC RATIOS 6 In the diagram below of right triangle KTW, KW = 6, KT = 5, and m KTW = 90. Evaluate: 0 + (n ) 5 n = Mrs. Hill asked her students to express the sum using sigma notation. Four different student answers were given. Which student answer is correct? 0 (k ) k = 0 (k ) k = 7 (k + ) k = 9 (k ) k = What is the measure of K, to the nearest minute? ' ' 55' 56' Express the sum using sigma notation.

25 7 Which ratio represents csc A in the diagram below? In the diagram below of right triangle JTM, JT =, JM = 6, and m JMT = 90. What is the value of cot J? A.M.-: RADIAN MEASURE 9 What is the radian measure of the smaller angle formed by the hands of a clock at 7 o clock? π π 5π 6 7π 6 50 What is the radian measure of an angle whose measure is 0? 7π 7π 6 7π 6 7π 5 What is the number of degrees in an angle whose radian measure is π? Find, to the nearest minute, the angle whose measure is.5 radians. 5 Find, to the nearest tenth of a degree, the angle whose measure is.5 radians.

26 A.A.60: UNIT CIRCLE 5 In which graph is θ coterminal with an angle of 70? 55 On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 0 and find the exact value of sin0. A.A.6, 66: DETERMINING TRIGONOMETRIC FUNCTIONS 56 The value of tan6 to the nearest ten-thousandth is If θ is an angle in standard position and its terminal side passes through the point (,), find the exact value of csc θ. A.A.6: USING INVERSE TRIGONOMETRIC FUNCTIONS 58 What is the principal value of cos ?

27 59 In the diagram below of a unit circle, the ordered pair, represents the point where the terminal side of θ intersects the unit circle. A.A.6: ARC LENGTH 6 A circle has a radius of inches. In inches, what is the length of the arc intercepted by a central angle of radians? π 8π 8 A.A.58: COFUNCTION AND RECIPROCAL TRIGONOMETRIC FUNCTIONS 6 If A is acute and tana =, then cot A = cot A = cot(90 A) = What is m θ? If sin 5 = A, then 8 sina = 5 8 sina = 8 5 cos A = 5 8 cos A = 8 5 A.A.57: REFERENCE ANGLES 6 Expressed as a function of a positive acute angle, cos( 05 ) is equal to cos 55 cos 55 sin 55 sin 55 cot(90 A) = 6 The expression sin θ + cos θ sin θ is equivalent to cos θ sin θ sec θ csc θ A.A.67: PROVING TRIGONOMETRIC IDENTITIES 65 Starting with sin A + cos A =, derive the formula tan A + = sec A. A.A.76: ANGLE SUM AND DIFFERENCE IDENTITIES 66 The expression cos x cos x + sinx sinx is equivalent to sinx sin7x cos x cos 7x

28 67 If tana = and sinb = 5 and angles A and B are in Quadrant I, find the value of tan(a + B). 68 Express as a single fraction the exact value of sin 75. A.A.77: DOUBLE AND HALF ANGLE IDENTITIES 69 The expression cos θ cos θ is equivalent to sin θ sin θ cos θ + cos θ 70 If sina = where 0 < A < 90, what is the value of sina? A.A.69: PROPERTIES OF TRIGONOMETRIC FUNCTIONS 7 What is the period of the function f(θ) = cos θ? π π π π 75 What is the period of the function y = sin x π? π 6π A.A.68: TRIGONOMETRIC EQUATIONS 7 What are the values of θ in the interval 0 θ < 60 that satisfy the equation tanθ = 0? 60º, 0º 7º, 5º 7º, 08º, 5º, 88º 60º, 0º, 0º, 00º 7 Solve the equation tanc = tanc algebraically for all values of C in the interval 0 C < Find all values of θ in the interval 0 θ < 60 that satisfy the equation sinθ = sinθ. 5

29 A.A.65, 70-7: GRAPHING TRIGONOMETRIC FUNCTIONS 77 Which equation is represented by the graph below? 76 Which graph represents one complete cycle of the equation y = sin πx? y = cot x y = csc x y = sec x y = tanx 78 Which equation is sketched in the diagram below? y = csc x y = sec x y = cot x y = tanx 6

30 79 Which graph represents the equation y = cos x? 80 Which graph shows y = cos x? 7

31 A.A.6: DOMAIN AND RANGE 8 The function f(x) = tanx is defined in such a way that f (x) is a function. What can be the domain of f(x)? {x 0 x π} {x 0 x π} x π < x < π x π < x < π A.A.7: USING TRIGONOMETRY TO FIND AREA 8 In ABC, m A = 0, b = 0, and c = 8. What is the area of ABC to the nearest square inch? The sides of a parallelogram measure 0 cm and 8 cm. One angle of the parallelogram measures 6 degrees. What is the area of the parallelogram, to the nearest square centimeter? A.A.75: LAW OF SINES-THE AMBIGUOUS CASE 86 How many distinct triangles can be formed if m A = 5, a = 0, and b =? 0 87 In ABC, m A = 7, a = 59., and c = 60.. What are the two possible values for m C, to the nearest tenth? 7.7 and and and and 68. A.A.7: LAW OF COSINES 88 In ABC, a = 5, b =, and c =, as shown in the diagram below. What is the m C, to the nearest degree? 8 Two sides of a parallelogram are feet and 0 feet. The measure of the angle between these sides is 57. Find the area of the parallelogram, to the nearest square foot. A.A.7: LAW OF SINES 85 In ABC, m A =, a =, and b = 0. Find the measures of the missing angles and side of ABC. Round each measure to the nearest tenth In ABC, a =, b = 5, and c = 7. What is m C?

32 90 In a triangle, two sides that measure 6 cm and 0 cm form an angle that measures 80. Find, to the nearest degree, the measure of the smallest angle in the triangle. 9 A circle shown in the diagram below has a center of ( 5,) and passes through point (,7). A.A.7: VECTORS 9 Two forces of 5 newtons and 85 newtons acting on a body form an angle of 55. Find the magnitude of the resultant force, to the nearest hundredth of a newton. Find the measure, to the nearest degree, of the angle formed between the resultant and the larger force. CONICS A.A.7, 9: EQUATIONS OF CIRCLES 9 The equation x + y x + 6y + = 0 is equivalent to (x ) + (y + ) = (x ) + (y + ) = 7 (x + ) + (y + ) = 7 (x + ) + (y + ) = 0 Write an equation that represents the circle. 95 Which equation represents the circle shown in the graph below that passes through the point (0, )? 9 Write an equation of the circle shown in the graph below. (x ) + (y + ) = 6 (x ) + (y + ) = 8 (x + ) + (y ) = 6 (x + ) + (y ) = 8 9