MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
|
|
- Lucas Potter
- 7 years ago
- Views:
Transcription
1 Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Tell in which quadrant or on what coordinate ais the point lies. 1) (, -19) A) Quadrant II B) Quadrant III Quadrant I D) Quadrant IV 1) Find the distance d(p1, P) between the points P1 and P. ) P1 = (6, ); P = (-, -6) A) 18 B) 7 D) 88 ) Solve the problem. ) Find all the points having an -coordinate of 9 whose distance from the point (, -) is 10. A) (9, -1), (9, 8) B) (9, 1), (9, -7) (9, ), (9, -) D) (9, 6), (9, -10) ) List the intercepts of the graph. ) 1 ) -π -π π π A) - π, 0, (, 0), π, 0 B) 0, - π, (0, ), 0, π - π, 0, (0, ), π, 0 D) 0, - π, (, 0), 0, π List the intercepts for the graph of the equation. ) = A) (0, -), (0, -), (16, 0) B) (0, ), (0, ), (16, 0) (-, 0), (-, 0), (0, 16) D) (, 0), (, 0), (0, 16) ) 1
2 Determine whether the graph of the equation is smmetric with respect to the -ais, the -ais, and/or the origin. 6) = 6) + 9 A) -ais B) -ais origin D) -ais, -ais, origin E) none Solve the problem. 7) Find the equation of a circle in standard form where C(6, -) and D(-, ) are endpoints of a diameter. A) ( + 1) + ( + 1) = B) ( - 1) + ( - 1) = ( + 1) + ( + 1) = 16 D) ( - 1) + ( - 1) = 16 7) Determine whether the relation represents a function. If it is a function, state the domain and range. 8) Alice snake Brad cat Carl dog 8) A) function domain: {Alice, Brad, Carl} range: {snake, cat, dog} B) function domain: {snake, cat, dog} range: {Alice, Brad, Carl} not a function 9) {(-, ), (-1, 0), (0, -1), (1, 0), (, 8)} A) function domain: {, 0, -1, 8} range: {-, -1, 0, 1, } B) function domain: {-, -1, 0, 1, } range: {, 0, -1, 8} not a function 9) Find the value for the function. 10) Find f() when f() = +. A) 6 B) 0 1 D) 10) Find the domain of the function. - 11) h() = - A) { 0} B) { -, 0, } { } D) all real numbers 11) 1) f() = 7 - A) { 7} B) { 7} { 7} D) { 7} 1)
3 Determine whether the graph is that of a function. If it is, use the graph to find its domain and range, the intercepts, if an, and an smmetr with respect to the -ais, the -ais, or the origin. 1) 1) 1 -π -π -π -π π π π π -1 A) function domain: { -π π} range: { -1 1} intercepts: (-π, 0), (- π, 0), (- π, 0), (- π, 0), (0, 0), ( π, 0), ( π, 0), ( π, 0), ( π, 0), (π, 0) smmetr: origin B) function domain: all real numbers range: { -1 1} intercepts: (-π, 0), (- π, 0), (- π, 0), (- π, 0), (0, 0), ( π, 0), ( π, 0), ( π, 0), ( π, 0), (π, 0) smmetr: origin function domain: { -1 1} range: { -π π} intercepts: (-π, 0), (- π, 0), (- π, 0), (- π, 0), (0, 0), ( π, 0), ( π, 0), ( π, 0), ( π, 0), (π, 0) smmetr: none D) not a function Answer the question about the given function. 1) Given the function f() = , is the point (, ) on the graph of f? A) Yes B) No 1) Determine algebraicall whether the function is even, odd, or neither. 1) f() = + A) even B) odd neither 1)
4 The graph of a function is given. Determine whether the function is increasing, decreasing, or constant on the given interval. 16) (-1, 0) 16) A) increasing B) constant decreasing Use a graphing utilit to graph the function over the indicated interval and approimate an local maima and local minima. Determine where the function is increasing and where it is decreasing. If necessar, round answers to two decimal places. 17) f() = - +, (-1, ) A) local maimum at (, -1) local minimum at (0, ) increasing on (-1, 0) decreasing on (0, ) local maimum at (, -1) local minimum at (0, ) increasing on (-1, 0) and (, ) decreasing on (0, ) B) local maimum at (0, ) local minimum at (, -1) increasing on (-1, 0) and (, ) decreasing on (0, ) D) local maimum at (0, ) local minimum at (, -1) increasing on (0, ) decreasing on (-1, 0) and (, ) 17) Using transformations, sketch the graph of the requested function. 18) The graph of a function f is illustrated. Use the graph of f as the first step toward graphing the function F(), where F() = f( + ) ) (-1, 1) - (-, -) - (, -)
5 A) B) (-, 1) (-, ) - (-, -) (1, -) - - (-, -1) (1, -) - D) (-, 0) (1, 0) - - (-, -) (-1, -) - (1, -) - (, -) Convert the angle to a decimal in degrees. Round the answer to two decimal places. 19) ʹ18ʹʹ A).9 B).9.98 D).88 19) 0) 8 1ʹʹʹ A) 8.18 B) D) 8.8 0) If s denotes the length of the arc of a circle of radius r subtended b a central angle θ, find the missing quantit. 1) r = 1 feet, s = 7 feet, θ =? 1) A) 7 B) 7 radians 1 radians D) 1 Convert the angle in degrees to radians. Epress the answer as multiple of π. ) 7 A) 6π 1 B) π 11 π 1 D) 1π )
6 Solve the problem. ) The Earth rotates about its pole once ever hours. The distance from the pole to a location on Earth 9 north latitude is about 98 miles. Therefore, a location on Earth at 9 north latitude is spinning on a circle of radius 98 miles. Compute the linear speed on the surface of the Earth at 9 north latitude. A) 680 mph B) 108 mph 68 mph D) 16, mph ) In the problem, t is a real number and P = (, ) is the point on the unit circle that corresponds to t. Find the eact value of the indicated trigonometric function of t. ) ( 8, ) Find tan t. ) 8 A) 8 B) 8 D) Find the eact value. Do not use a calculator. ) sin (- π ) ) A) 1 B) -1 0 D) undefined Find the eact value of the epression if θ =. Do not use a calculator. 6) g(θ) = sin θ Find [g(θ)]. A) 1 B) - D) 6) Find the eact value of the epression. Do not use a calculator. 7) sin π - cos π 6 7) A) 0 B) - 1 D) 1 Find the eact value of the epression if θ = 0. Do not use a calculator. 8) g(θ) = cos θ Find g(θ). A) 1 B) 1 D) 8) Find the eact value of the epression. Do not use a calculator. 9) cos π + tan π 9) A) + 1 B) D) + A point on the terminal side of an angle θ is given. Find the eact value of the indicated trigonometric function of θ. 0) (-, -1) Find cos θ. 0) A) - B) 1 D)
7 Solve the problem. 1) For what numbers θ is f(θ) = tan θ not defined? A) odd multiples of π (90 ) B) all real numbers 1) integral multiples of π (180 ) D) odd multiples of π (180 ) Use the fact that the trigonometric functions are periodic to find the eact value of the epression. Do not use a calculator. ) cot 70 ) A) B) - - D) ) sin 11π ) A) - B) - 1 D) -1 Name the quadrant in which the angle θ lies. ) sin θ > 0, cos θ < 0 A) I B) II III D) IV ) In the problem, sin θ and cos θ are given. Find the eact value of the indicated trigonometric function. ) sin θ = 1, cos θ = 1 Find csc θ. A) B) D) 1 1 ) Find the eact value of the indicated trigonometric function of θ. 6) cos θ = 8 17, π < θ < π Find cot θ. A) B) D) ) Use the even-odd properties to find the eact value of the epression. Do not use a calculator. 7) sin (-10 ) A) B) 1-1 D) - 7) Without graphing the function, determine its amplitude or period as requested. 8) = - sin Find the amplitude. A) π B) -π D) π 8) 9) = sin Find the period. 9) A) π B) 1 D) π 7
8 Answer the question. 0) Which one of the equations below matches the graph? 0) A) = cos B) = cos 1 = - sin D) = sin 1 Find an equation for the graph. 1) 1) 1 -π -π -1 π π A) = cos 1 B) = cos () = cos () D) = cos 1 Solve the problem. ) For the equation = - 1 period. cos( - π), identif (i) the amplitude, (ii) the phase shift, and (iii) the ) A) (i) 1 (ii) π (iii) π B) (i) (ii) π (iii) π (i) 1 (ii) π (iii) π D) (i) (ii) π (iii) π Find the eact value of the epression. ) cos-1 - ) A) π 6 B) π π 6 D) π 8
9 ) tan-1 ) A) 7π 6 B) π π D) π 6 Find the eact value of the epression. Do not use a calculator. ) sin [sin-1 (0.)] A) B) D).0 ) 6) sin-1 sin π 7 6) A) 7 π B) π 7 π 7 D) 7 π Use a calculator to find the value of the epression rounded to two decimal places. 7) sin ) A) 8.8 B) D) 1. Find the eact solution of the equation. 8) cos-1 = π 8) A) = 0 B) = π = π D) = 1 Find the eact value of the epression. 9) sin (tan-1 ) 9) A) B) D) 0) cos sin-1 0) A) - B) - 1 D) 1) cot-1-1) A) π 6 B) π 6 π D) π ) sec-1 (-) ) A) - π B) π π D) - π Use a calculator to find the value of the epression in radian measure rounded to two decimal places. ) cot-1 A) 1.7 B) D) 0.0 ) 9
10 Complete the identit. ) (sin θ + cos θ) 1 + sin θ cos θ =? ) A) 1 - sin θ B) -sec θ 1 D) 0 ) cos θ - cos θ sin θ =? A) sec θ B) cos θ tan θ D) sin θ ) Find the eact value of the epression. 6) sin - 11π 1 6) A) B) D) 6 - Find the eact value under the given conditions. 7) sin α = 1 9, 0 < α < π 1 ; cos β = 1, 0 < β < π Find cos (α + β). 7) A) 77 B) D) 77 8) tan α = 7, π < α < π ; cos β = - 1 1, π < β < π Find sin (α + β). 8) A) B) 0 D) - 6 Find the eact value of the epression. 9) cos tan-1 - sin -1 9) A) B) 6 D) 1 Use the information given about the angle θ, 0 θ π, to find the eact value of the indicated trigonometric function. 60) cos θ = 1 17, π < θ < π Find sin (θ). 60) A) B) D) ) csc θ = -, tan θ > 0 Find cos (θ). 61) A) 1 9 B) D) 9 6) cos θ = -, π < θ < π Find cos θ. 6) A) B) D)
11 6) sin θ = -, π < θ < π Find sin θ. 6) A) B) D) - Epress the product as a sum containing onl sines or cosines. 6) sin (θ) cos (θ) 6) A) 1 [sin (7θ) + sin (θ)] B) sin cos (10θ ) 1 [sin (7θ) + cos (θ)] D) 1 [cos (7θ) - cos (θ)] Epress the sum or difference as a product of sines and/or cosines. 6) cos (6θ) + cos (θ) A) cos (θ) B) cos (θ) sin θ sin (θ) sin θ D) cos (θ) cos θ 6) Solve the equation on the interval 0 θ < π. 66) sin (θ) = 66) A) π, π B) 0 0, π, π D) π 1, π 6, π, 7π 1, 7π 6, 1π 1, π, 19π 1 Solve the equation. Give a general formula for all the solutions. 67) cos (θ) = 67) A) θ = π 8 + kπ, θ = 7π 8 + kπ B) θ = π 8 + kπ, θ = 7π 8 + kπ θ = π + kπ, θ = π + kπ D) θ = π + kπ, θ = π + kπ Solve the equation on the interval 0 θ < π. 68) sin θ = sin θ 68) A) π 6, π 6 B) π, π, π, π π, π D) 0, π, π 6, π 6 69) sin θ - cos θ = 0 69) A) π, π 6 B) π, π π, π, π, 7π D) π 11
12 Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give eact answers with rational denominators. 70) Find sin B when b = and c =. 70) A) B) D) 7 7 Solve the right triangle using the information given. Round answers to two decimal places, if necessar. 71) a =, b = 6; Find c, α, and β. A) c = 7.81 B) c =.7 α = 9.81 α = 9.81 β = 0.19 β = 0.19 c = 7.81 α = 0.81 β = 9.19 D) c =.7 α = 0.81 β = ) Solve the triangle. 7) β = 70, γ = 60, b = A) α = 0, c =.69, a =.6 B) α = 0, c =.6, a =.69 α = 0, c =.6, a =.69 D) α = 0, c =.69, a =.6 7) Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve an triangle(s) that results. 7) a = 7, b = 9, β = 9 7) A) one triangle α = 76.01, γ =.99, c = 7.60 B) one triangle α =.9, γ = 9.06, c = two triangles α1 = 76.01, γ1 =.99, c1 = 7.60 or α = 10.99, γ = 7.01, c = 1.1 D) no triangle 7) a = 1, b = 8, β = 10 A) two triangles α1 = 1.1, γ1 = 1.9, c1 = 19. or α = 16.9, γ =.1, c =.1 one triangle α = 1.1, γ = 1.9, c = 19. B) one triangle α = 16.9, γ =.1, c =.1 D) no triangle 7) Solve the triangle. 7) b =, c = 6, α = 80 A) a = 8.11, β =.8, γ = 6. B) a = 6.11, β = 6., γ =.8 a = 7.11, β =.8, γ = 6. D) a = 7.11, β = 6., γ =.8 7) 1
13 Solve the triangle. Find the angles α and β first. 76) a = 7, b = 1, c = 17 A) α = 0., β =.8, γ = 11.9 B) α =., β =.8, γ = 11.9 α =., β =.8, γ = 11.9 D) no triangle 76) Find the area of the triangle. If necessar, round the answer to two decimal places. 77) α = 0, b = 1, c = 6 A) 18 B) D) ) 78) a = 1, b =, c = 6 A) B) D) ) Match the point in polar coordinates with either A, B, C, or D on the graph. 79) -, π 79) A 1 B r -1 - D C A) A B) B C D) D The polar coordinates of a point are given. Find the rectangular coordinates of the point. 80) 7, π 80) A) - 7, 7 B) 7, -7 7, 7 D) - 7, -7 The rectangular coordinates of a point are given. Find polar coordinates for the point. 81) (-, ) 81) A), - π B) -, - π, π D) -, π Write the comple number in polar form. Epress the argument in degrees, rounded to the nearest tenth, if necessar. 8) - i 8) A) (cos 0 + i sin 0 ) B) (cos 00 + i sin 00 ) (cos 00 + i sin 00 ) D) (cos 0 + i sin 0 ) 1
14 Solve the problem. Leave our answer in polar form. 8) z = 10(cos + i sin ) w = (cos 1 + i sin 1 ) Find zw. A) (cos 60 + i sin 60 ) B) 0(cos 60 + i sin 60 ) 0(cos 0 + i sin 0 ) D) (cos 0 + i sin 0 ) 8) Write the epression in the standard form a + bi. 8) cos π 6 + i sin π 6 8) A) i B) i i D) i Find all the comple roots. Leave our answers in polar form with the argument in degrees. 8) The comple fourth roots of -16 8) A) (cos + i sin ), (cos 1 + i sin 1 ), (cos + i sin ), (cos 1 + i sin 1 ) B) (cos 90 + i sin 90 ), (cos i sin 180 ), (cos 70 + i sin 70 ), (cos 60 + i sin 60 ) 16(cos + i sin ), 16(cos 1 + i sin 1 ), 16(cos + i sin ), 16(cos 1 + i sin 1 ) D) (cos + i sin ), (cos 1 + i sin 1 ), (cos + i sin ), 16(cos 1 + i sin 1 ) The vector v has initial position P and terminal point Q. Write v in the form ai +bj; that is, find its position vector. 86) P = (, ); Q = (-1, -) 86) A) v = 7i + 6j B) v = 6i + 7j v = -6i - 7j D) v = -7i - 6j Solve the problem. 87) If u = -10i - j and v = -i + 7j, find u - v. A) -9i + j B) -10i + j -1i + j D) -8i - 9j 87) 88) If v = 6i - 8j, find v. A) 1 B) D) 10 88) Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive -ais. 89) v = 1, α = 10 89) A) v = 1-1 i - 1 j B) v = 1-1 i - 1 j v = 1 - i - 1 j D) v = 1 i - 1 j 1
15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 90) An audio speaker that weighs 0 pounds hangs from the ceiling of a restaurant from two cables as shown in the figure. To two decimal places, what is the tension in the two cables? 90) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the dot product v w. 91) v = 1i + j, w = -1i - j A) -18 B) D) ) Find the angle between v and w. Round our answer to one decimal place, if necessar. 9) v = -i + 7j, w = -6i - j A) 90.9 B) D) 0.7 9) Solve the problem. 9) Which of the following vectors is parallel to v = -10i - 8j? A) w = 0i + 16j B) w = i - j w = -0i + j D) w = i + j 9) State whether the vectors are parallel, orthogonal, or neither. 9) v = i + j, w = 6i + 8j A) Parallel B) Orthogonal Neither 9) 9) v = i + j, w = i - j A) Orthogonal B) Parallel Neither 9) Solve the problem. Round our answer to the nearest tenth. 96) A wagon is pulled horizontall b eerting a force of 60 pounds on the handle at an angle of to the horizontal. How much work is done in moving the wagon 0 feet? ʺ. A) ft-lb B) ft-lb ft-lb D) ft-lb 96) 1
16 Answer Ke Testname: TRIG FINAL 1) D ) A ) D ) C ) C 6) C 7) B 8) C 9) B 10) A 11) B 1) C 1) A 1) B 1) B 16) C 17) B 18) C 19) B 0) B 1) C ) C ) A ) C ) B 6) A 7) A 8) B 9) C 0) A 1) A ) D ) A ) B ) A 6) B 7) D 8) B 9) D 0) B 1) D ) A ) C ) D ) B 6) C 7) C 8) A 9) B 0) D 16
17 Answer Ke Testname: TRIG FINAL 1) B ) C ) D ) C ) B 6) B 7) C 8) D 9) C 60) D 61) A 6) B 6) B 6) A 6) D 66) D 67) B 68) D 69) C 70) A 71) A 7) A 7) B 7) A 7) C 76) B 77) A 78) D 79) D 80) A 81) C 8) D 8) B 8) B 8) D 86) C 87) D 88) C 89) B 90) Tension in right cable:.90 lb; tension in left cable: 1.9 lb 91) C 9) B 9) A 9) A 9) A 96) A 17
135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationFind the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
More informationSection 5-9 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationTrigonometry LESSON ONE - Degrees and Radians Lesson Notes
210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationThe Circular Functions and Their Graphs
LIALMC_78.QXP // : AM Page 5 The Circular Functions and Their Graphs In August, the planet Mars passed closer to Earth than it had in almost, ears. Like Earth, Mars rotates on its ais and thus has das
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationTrigonometry Hard Problems
Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.
More informationStart Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES
COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided
More informationSECTION 7-4 Algebraic Vectors
7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More information( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:
Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More information2.5 Library of Functions; Piecewise-defined Functions
SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationPROBLEM 2.9. sin 75 sin 65. R = 665 lb. sin 75 sin 40
POBLEM 2.9 A telephone cable is clamped at A to the pole AB. Knowing that the tension in the right-hand portion of the cable is T 2 1000 lb, determine b trigonometr (a) the required tension T 1 in the
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More information8-3 Dot Products and Vector Projections
8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationDear Accelerated Pre-Calculus Student:
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
More informationMCA Formula Review Packet
MCA Formula Review Packet 1 3 4 5 6 7 The MCA-II / BHS Math Plan Page 1 of 15 Copyright 005 by Claude Paradis 8 9 10 1 11 13 14 15 16 17 18 19 0 1 3 4 5 6 7 30 8 9 The MCA-II / BHS Math Plan Page of 15
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationChapter 5: Trigonometric Functions of Angles
Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationSection 10.4 Vectors
Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationRight Triangle Trigonometry
Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned
More information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationSOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions
More informationMATH 185 CHAPTER 2 REVIEW
NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationLaw of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
More informationex) What is the component form of the vector shown in the picture above?
Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationCircles - Past Edexcel Exam Questions
ircles - Past Edecel Eam Questions 1. The points A and B have coordinates (5,-1) and (13,11) respectivel. (a) find the coordinates of the mid-point of AB. [2] Given that AB is a diameter of the circle,
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationEvaluating trigonometric functions
MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11.
SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Tuesday, August 16, 2005 8:30 to 11:30 a.m.
MATHEMATICS B The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Tuesday, August 16, 2005 8:30 to 11:30 a.m., only Print Your Name: Print Your School's Name: Print your
More informationSelf-Paced Study Guide in Trigonometry. March 31, 2011
Self-Paced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the Self-Paced Review Module Trigonometry Self-Paced Review Module 4.1 Right Triangles..........................
More informationFourth Grade Math Standards and "I Can Statements"
Fourth Grade Math Standards and "I Can Statements" Standard - CC.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and
More information1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
More informationUnit 11 Additional Topics in Trigonometry - Classwork
Unit 11 Additional Topics in Trigonometry - Classwork In geometry and physics, concepts such as temperature, mass, time, length, area, and volume can be quantified with a single real number. These are
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationSection 1-4 Functions: Graphs and Properties
44 1 FUNCTIONS AND GRAPHS I(r). 2.7r where r represents R & D ependitures. (A) Complete the following table. Round values of I(r) to one decimal place. r (R & D) Net income I(r).66 1.2.7 1..8 1.8.99 2.1
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationModule 8 Lesson 4: Applications of Vectors
Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationMidterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More information