trigonometry reproducibles pdf

www.EngineeringBooksPDF.com Trigonometry Reproducibles MP3510 Trigonometry Author: Marilyn Occhiogrosso Editor: Howard Brenner Project Director: Kathleen Coleman ISBN: 0-7877-0631-0 Copyright © 2007 Milliken Publishing Company 3190 Rider Trail South Earth City, MO 63045 www.millikenpub.com Printed in the USA All rights reserved The purchase of this book entitles the individual purchaser to reproduce copies by duplicating master or by any photocopy process for single classroom use The reproduction of any part of this book for commercial resale or for use by an entire school or school system is strictly prohibited Storage of any part of this book in any type of electronic retrieval system is prohibited unless purchaser receives written authorization from the publisher Milliken Publishing Company www.EngineeringBooksPDF.com Earth City, Missouri Table of Contents Measures of Angles Basic Trigonometric Identities 25 Arc Length Sum and Difference Identities 26 Trigonometric Functions of Acute Angles Double-Angle Identities 27 Half-Angle Identities 28 Applying Trigonometry in Right Triangles Redefining the Trigonometric Functions The Ambiguous Case 29 The Law of Sines 30 The Law of Cosines 31 Using One Function Value to Find Others Representing Trigonometric Functions as Line Segments Area of a Triangle 32 Parallelogram of Forces 33 Navigation Problems 34 Function Values of Quadrantal Angles 10 Literal Triangle Problems 35 Function Values of 30°, 45°, 60° 11 Polar Coordinates 36 Function Values of Angles of Any Size 12 Polar Graphs 37-38 Representing One Trigonometric Function in Terms of Another 13 Trigonometric Form of Complex Numbers 39 Inverse Trigonometric Notation 14 Solving First-Degree Trigonometric Equations 15 More Equations with First-Degree Trigonometric Functions 16 Solving Second-Degree Trigonometric Equations 17-18 Multiplication and Division in Trigonometric Form 40 Assessment A: Trigonometric Functions 41 Assessment B: Graphs of Trigonometric Functions 42 Graphs of Sine and Cosine Functions 19-20 Assessment C: Trigonometric Equations and Identities 43 Graphs of Other Trigonometric Functions 21-22 Assessment D: Solving Triangles 44 Graphing Systems of Trigonometric Equations 23 Answer Key 45-48 Graphs of Compound Trigonometric Functions 24 www.EngineeringBooksPDF.com Measures of Angles Name _ Remember An angle is formed by two rays with a common endpoint The size of an angle is the amount of rotation between its two rays Counterclockwise rotation is positive Clockwise rotation is negative Units for measuring rotation are revolution, degree, radian, and grad (gradian or gradient) Ter lS ide θ Initial Side Conversion Formulas: Initial Side −θ revolution = 360° π radians = 180° 100 grads = 90° Example: Convert 60° to radians a Ter a lS 60° x ide π radians π = radians 180° Play “odd measure out.” Cross through the measure in each row that is not equivalent to the other three 1 revolution 360˚ π radians revolution 12 133 grads 210° π radians 66 grads 60° 2π radians 233 400 grads revolution 7π radians grads 120° revolution 12 4π radians revolutions 720° 700 grads 1,200 grads 1,080° revolutions radians revolution 2π radians ⎛ 540 ⎞ ° ⎜⎝ π ⎟⎠ π grad 600 © Milliken Publishing Company www.EngineeringBooksPDF.com MP3510 Arc Length Name _ Remember s The linear measure, s, of an arc of a circle is related to the radian measure of its central angle, θ, and the radius, r θ r s=θr Example: Find the length of the arc that subtends an angle of 40° in a circle whose radius is 18 inches Answer to the nearest tenth of an inch π radians 2π = radians 180° Convert the angle measure from degrees to radians 40° x Substitute the radian value for θ and the value of the radius into the arc-length formula s = θr 2π s= x 18 Substitute for π, and evaluate s≈ 2(3.14) x 18 ≈ 12.56 ≈ 12.6 inches Find the indicated measures Then use the answer code to complete the seven-word sentence below In a circle with radius 12 centimeters, find the length of an arc intercepted by a central angle of 45° A circle has a radius of feet Find the radian measure of a central angle that intercepts an arc length of 12 feet In a circle, a central angle of 30° intercepts an arc of 23.5 inches Find the length of the radius The length of a pendulum is 18 inches Find the distance through which the tip of the pendulum travels when the pendulum turns through an arc of 2.5 radians The diameter of a wheel is 48 inches Find the number of degrees through which a point on the circumference turns when the wheel moves a distance of feet _ _ I _ _ I _ _ _ _ _ _ _ _ © Milliken Publishing Company 28.7° theta 57.3° pi 44.9 in could 45 in calculate 2π would wish 0.5 recalculate 3π cm how cm think Count the letters in each of the seven words Use the count to fill in the spaces below, revealing the values of the first seven digits of π www.EngineeringBooksPDF.com MP3510 Trigonometric Functions of Acute Angles Name Remember There are trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) Cofunctions: sine and cosine, tangent and cotangent, secant and cosecant Reciprocal Functions: (sine, cosecant), (cosine, secant), (tangent, cotangent) In a right triangle, the hypotenuse is always opposite the right angle In right triangle ABC, legs a and b are named with respect to acute angle A sin A = c b cos A = , sec A = b c tan A = A c a , csc A = a c hypotenuse c adjacent b a b , cot A = b a C Across sin A = _ B relationship between the acute angles of a right triangle _ A = cot B 10 opposite adjacent 11 hypotenuse adjacent 12 opposite hypotenuse side opposite right angle _ A = tan A © Milliken Publishing Company sin A = _ A Down hypotenuse opposite B opposite a 10 11 cos A adjacent hypotenuse _ A = 12 product of two reciprocals adjacent opposite www.EngineeringBooksPDF.com MP3510 Applying Trigonometry in Right Triangles Name _ Example C Use the information in the diagram to find BC and CD BC is a side of right ᭝   ABC tan ∠CAB = opposite BC = adjacent AB D BC tan 50° = 28 BC = 28(tan 50°) BC ≈ 28(1.1918) BC ≈ 33.4 ft 50° 32° A Use right ᭝ABD to find BD BD tan 32° = 28 BD = 28(tan 32°) BD ≈ 28(0.6249) ≈ 17.5 ft B 28 ft Subtract to get CD CD = BC – BD CD ≈ 33.4 – 17.5 CD ≈ 15.9 ft Find the measures of the lettered segments Use the answer code to fill in the blanks that follow for a-h, and reveal the reason the Ancient Greeks developed trigonometry 41° 85 m b 16 ft 20° d 35° c a e 28° 100 ft 60 ft g f 121.4 T h 37° 35° 90 ft 18 A 120 m 122.1 O 20 N 24 S 112.9 D 112.1 R 154.2 Y 22 P 63.8 M a b c d e f e g h © Milliken Publishing Company www.EngineeringBooksPDF.com MP3510 Name _ Redefining the Trigonometric Functions Remember In the coordinate plane, if P(x, y) is a point on a circle with center at the origin and radius r, then: y x y sin θ = , cos θ = , tan θ = r r x y The function values will be positive or negative depending on the quadrant for the terminal side of θ y QI: All positive QII: Sin positive QIII: Tan positive QIV: Cos positive II I S A P(x, y) r θ x x y x T C III IV Reciprocals are positive where the basic functions are positive Use the information in each cell to determine the quadrant for angle θ sinθ > 0, tanθ > θ is in Q _ cosθ > 0, sinθ < θ is in Q _ sinθ > 0, cosθ < θ is in Q _ cscθ > 0, cosθ < θ is in Q _ cotθ < 0, secθ > θ is in Q _ cotθ > 0, sinθ < θ is in Q _ cotθ < 0, cosθ > θ is in Q _ secθ < 0, tanθ > θ is in Q _ cotθ < 0, cscθ > θ is in Q _ Create a table in which the angles in a given row are in the same quadrant Cells in a given row must all be different Do not repeat cells from the table above θ is in Q I θ is in Q I θ is in Q I θ is in Q I θ is in Q II θ is in Q II θ is in Q II θ is in Q II θ is in Q III θ is in Q III θ is in Q III θ is in Q III θ is in Q IV θ is in Q IV θ is in Q IV θ is in Q IV © Milliken Publishing Company www.EngineeringBooksPDF.com MP3510 Name _ Using One Function Value to Find Others Remember If you know the quadrant in which an angle lies and the value of one of its functions, then you can find the other function values for that angle and sinθ < 0, find cosθ Since tanθ > and sinθ < 0, θ lies in Quadrant III y Example: If tanθ = In Quadrant III, both x and y are negative –4 y − Use y = – and x = – tanθ = = = x −4 Use the Pythagorean Relation to find r θ x –3 r = ( − 3) + ( − 4) r = 25, use r > 0,  r = x −4 Thus: cosθ = = =− 5 r and cotθ > 0, then secθ = and cosθ < 0, then cotθ = If sinθ = − If cotθ = 15 and sinθ < 0, then secθ = If tanθ = If tanθ = and cosθ < 0, If sinθ = If sinθ = and secθ > 0, then cscθ = and cotθ < 0, 17 then secθ = then cscθ = and cotθ > 0, then sinθ = If cotθ = − and secθ < 0, If secθ = then sinθ = and cscθ > 0, then tanθ = If secθ = − Connect your answers in the order of the problem numbers You should be connecting each dot to every other dot without drawing the same line segment twice © Milliken Publishing Company − 17 15 − www.EngineeringBooksPDF.com − MP3510 Representing Trigonometric Functions as Line Segments Name _ Remember Using a circle with radius 1, called a unit circle, the trigonometric functions can be represented as line segments y Example: The diagram shows a unit circle, with center at the origin ៮៮៮៮ and ៮៮៮៬ PQ TF are each perpendicular to (0,1) the x-axis Find segments to represent sin θ, cos θ, and tan θ, respectively sinθ = PQ PQ ៮៮៮៮ ; PQ represents sin θ = OP cos θ = tanθ = O (–1,0) OQ OQ ៮៮៮៮ represents cos θ ; OQ = OP T P θ F Q x (1,0) (0,–1) TF TF ៮៮៮៮represents tan θ ; TF = OF Select a line segment to represent each of the indicated functions The diagram shows a unit circle, with ៮៮៮៮៮៮ and TBT´ ៮៮៮៮៮៮ are center at the origin PAP´ ៮៮៮៮ is each perpendicular to the x-axis QR perpendicular to the y-axis T y (0,1) R Q P a sinθ is represented by b tanθ is represented by c cosθ is represented by θ O –θ (–1,0) A B (1,0) x d cos(–θ) is represented by e sin(–θ) is represented by P´ f tan(–θ) is represented by (0,–1) g cotθ is represented by h secθ is represented by T´ ៮៮៮៮ P´A ៮៮៮៮ PA ៮៮៮៮ AB ៮៮៮៮ TB ៮៮៮៮ OA ៮៮៮៮ T´B ៮៮៮៮ OA ៮៮៮៮ RQ ៮៮៮៮ OQ ៮៮៮៮ OT E S O Y M T M R A Y Use the answer code to reveal an attribute of the unit circle a b c d e f g h © Milliken Publishing Company www.EngineeringBooksPDF.com MP3510 Name _ Navigation Problems Remember N Bearings (directions) are given by course angles between 0° and 360°, beginning at North and moving clockwise Equivalent bearings are named by an angle measure between two compass directions 0° 60° E W 270° 300° As shown in the diagram, the course angle 300° is equivalent to the bearing N 60° W 90° 180° S Solve each problem Match your answers to those in the code below b A ship leaves port P on a bearing of 32° and travels miles Then the ship turns due East and travels miles How far is the ship from port P? a Two ranger stations, X and Y, are on an east-west line 75 miles apart A fire is located at point F, bearing N 55° E of station X and N 10° E of station Y How far is the fire from station X? N mi F N mi 0° ? ? 32° 55° X 10° 75 mi Y P c Two ships leave port at the same time, one in a direction S 70° E (110°) at a rate of 18 miles per hour and the other in a direction S 85° W (265°) at a rate of 21 miles per hour How far apart are the ships at the end of hour? d At PM, a ship sailed from port bearing N 71° E at a speed of 15 mph At PM, a second ship left the same port bearing S 49° E at 20 mph How far apart are the ships at PM? e A game warden at tower A sights an injured zebra at a bearing of 295° A warden in tower B, which is located 45 miles at a bearing of 45° from tower A, sees the same zebra at a bearing of 255° How far from tower A is the injured zebra? A ship leaves a port and sails 30 miles due East Then the ship turns and sails 40 miles in the direction N 30° E, and drops anchor A second ship leaves the same port to rendezvous with the first ship f What course should the second ship take from the port? g What distance will the second ship travel? 13 mi Z N 35° E N 104 mi A N 55° E T 38 mi I 43 M 61 mi H 35 mi U Use the code to reveal the term applied to bearings that are taken with respect to angles of the sun These direction lines are associated with sundials and navigation compasses _ _ _ _ _ _ _ a b c d e f g © Milliken Publishing Company 34 www.EngineeringBooksPDF.com MP3510 Name _ Literal Triangle Problems Example A AD ⊥ BCD , ∠ABD = θ , ∠ACD = 2θ Verify that : tanθ = sin 2θ + cos θ 2θ θ Since ∠ACD is an exterior angle for ᭝ABC, B C m∠ACD = m∠ABC + m∠BAC 2θ = θ + m∠BAC θ = m∠BAC So, ABC is isosceles AD CD In ᭝ACD : sin 2θ = = AD and cos 2θ = = CD 1 D then: A θ θ sin 2θ AD AD In ᭝ABD : tanθ = = = BD BC + CD + cos θ 2θ B C D Show the equivalence of the indicated expressions Transfer the code letters from the shaded boxes to the blanks at the bottom of the page a PQ SR , QS ⊥ PS , QR ⊥ SR , PQ = ∠QPS = ∠RQS = θ Q θ V V θ P STP ⊥ PQ , RQ ⊥ PQ , RT ⊥ STP ∠RPQ = α , ∠RST = β , SR = S β R T R S SR = sin2 θ c b P α RQ = tan α sin β B QR ⊥ PSR , ∠QPR = α , ∠QSR = β , QR = h, PS = x, SR = y d Q 2θ h P x = h (cot α − cot β ) R R P θ PR = sinθ R Reveal the type of thinking you used in this activity © Milliken Publishing Company S QS ⊥ PS , ∠RPS = θ ∠PQS = 2θ , PQ = Q β α x S y Q S C A T A T a b c d 35 www.EngineeringBooksPDF.com MP3510 Name _ Polar Coordinates Remember In rectangular coordinates, a point is located by the coordinates (x, y), representing a horizontal distance from the origin and vertical distance from the x-axis In polar coordinates, a point is located by the coordinates (r, θ), representing a directed angle from the polar axis (positive portion of the x-axis) and a distance on the terminal side of that angle To plot points on a polar grid, first locate the ray for the directed angle Then move a directed distance r from the pole on that angular ray If r > 0, move on the ray If r < 0, move on the opposite ray y P(x, y) or (r, θ) Origin or Pole r θ y x-axis or Polar Axis x 30° P(3, 30°) Q(-2, 30°) Plot these points on the polar grid Use: 2 ≈ 2.8 and ≈ 3.5 A(0, 0) B(2, 30°) 105° 75° 120° C( 2, 45°) 60° 135° D( 3, 60°) E(4, 90°) 90° 45° 30° 150° F( 3, 120°) G( 2, 135°) H(2, 150°) 15° 165° 180° 0° I(0, 180°) J(–2, 210°) K( 2, 225°) L(–4, 270°) M( −2 , 300°) 345° 195° 330° 210° 315° 225° 300° 240° 255° 270° N(–2, 330°) 285° O(0, 360°) Use a smooth curve to connect your points in alphabetical order Make a conjecture as to the type of curve you have drawn Tell how you might verify your conjecture © Milliken Publishing Company 36 www.EngineeringBooksPDF.com MP3510 Polar Graphs Name _ Remember To graph a polar equation, r = f (θ ), make a table of values Choose values of θ, and calculate corresponding values of r Example: Sketch the graph of r = + cosθ cosθ cosθ r = + cosθ 0˚ 30˚ 60˚ 90˚ 120˚ 150˚ 180˚ 210˚ 240˚ 270˚ 300˚ 330˚ 360˚ 0.87 0.5 –0.5 –0.87 –1 –0.87 –0.5 0.5 0.87 1.74 –1 –1.74 –2 –1.74 –1 1.74 2.74 –0.74 –1 –0.74 2.74 90° 60° 120° 150° X 30° X X X X X 180° X3 0° X θ X X 210° 240° 330° 300° 270° A graph of the form r = a ± b cosθ or r = a ± b sin is called a limaỗon (French for snail) Another name is cardiod (for its heart shape) Depending on the ratio of a to b, the curve will have a “loop,” or not 90° 90° 180° 0° 270° a b 180° 0° 270°   <  1 90° 90° a b 180° 0° 270°   =  1 180° 0° 270° a 1  <     <   b a b  ≥ 2 The rose curves have equations of the form r = a cosnθ or r = a sinnθ The number of petals depends on the value of n If n is odd, there are n petals If n is even, there are 2n petals 90° 180° 90° 90° 0° 180° 180° 0° 90° 0° 180° 0° 270° 270° 270° 270° n=3 n=5 n=2 n=4 continued on pg 38 © Milliken Publishing Company 37 www.EngineeringBooksPDF.com MP3510 Name _ Polar Graphs (contd.) Match each of the graphs below with one of the following equations On the lines under each graph, write your choice of equation and the code letter given for that equation a r = sinθ R b r = sin 3θ A c r = – cosθ I d r = E e θ = 30° P f r = cos 2θ N g r2 = 16 cos 2θ S h θ = 60° M i r = – sinθ C j r= K sinθ k r= l r= 90° cos 2θ E cos θ 180° T cos θ + sinθ 90° 0° 180° 270° 270° 90° 180° 0° 180° 90° 0° 270° 180° 90° 0° 90° 270° 270° 90° 0° 90° 180° 0° 180° 180° 0° 270° 270° 270° 0° Use the code to reveal the name of the curve that is used as the symbol for infinity L _ _ _ _ _ _ _ _ _ _ © Milliken Publishing Company 38 www.EngineeringBooksPDF.com MP3510 Trigonometric Form of Complex Numbers Name _ Remember When θ is an angle in standard position with the point (x, y) on its terminal side, then: r = x + y , sinθ = y , cos θ = x , tanθ = y yi x r r standard form trigonometric form r x + yi r(cosθ + i sinθ) or r cisθ θ r is the modulus; θ is the argument x •(x, y) y x Example: Write the complex number −2 − 3i in trigonometric form ( Find r r = x + y → r = (−2)2 + −2 Find θ tanθ = y = x −2 −2 = ) = + 12 = 16 So, r = So, the reference angle = 60˚, with tanθ > Since y < 0, sinθ < 0; so θ is in QIII Thus, θ = 240° Therefore: −2 − 3i = r(cosθ + i sinθ) = 4(cos 240° + i sin 240°) or cis 240° 3 cis 315° i 4−4 i x = r cosθ 2 cis π © Milliken Publishing Company ( −i 11π 4 r cis θ ) 39 www.EngineeringBooksPDF.com cis 0° (r, θ) r cis 120° sinθ = y i + −5 5π +i + 2i cis cos θ = i i cis – 3i r (cosθ + i sinθ) − − 3 − + cis 120° i y = r sinθ (x, y) 8i r cis θ cis 30° x + yi − + 5π cis cis 300° r 4 − x cis 135° ⎛ π π⎞ ⎜ cos + i sin ⎟ 2⎠ ⎝ x2 + y2 = r 3 ✄ Cut the big square into small squares Create a new big square by matching equivalent expressions Along the perimeter of your square, opposite expressions should be equivalent r = x2 + y2 MP3510 Name _ Multiplication and Division in Trigonometric Form Remember To multiply: Multiply the moduli, and add the arguments r1 (cosθ + i sinθ ) i r2 (cosθ + i sinθ ) = r1r2 [cos(θ + θ ) + i sin(θ + θ )] To divide (not by 0): Divide the moduli, and subtract the arguments r1 (cosθ + i sinθ ) r1 = [cos(θ − θ ) + i sin(θ − θ )] r2 (cosθ + i sinθ ) r2 Multiply: (cos 40° + i sin 40°) ⋅ 2(cos 20° + i sin 20°) Write the answer in standard form Multiply the moduli and add the arguments 2 (cos 60° + i sin 60°) Example: ⎡1 ⎛ = 2⎢ +i ⎜ ⎢⎣ ⎝ Substitute the function values 3⎞⎤ ⎟⎥ = + 6i ⎠ ⎥⎦ Perform the indicated operations Write your answers in standard form a 4(cos 25° + i sin 25°) ⋅ (cos 20° + i sin 20°) b 8(cos 125° + i sin125°) ÷ 2(cos 35° + i sin 35°) c (cos 95° + i sin 95°) ⋅ (cos 25° + i sin 25°) d (cos 165° + i sin 165°) ÷ e 2 (cos 80° + i sin 80°) f ⋅ (cos15° + i sin 15°) (cos100° + i sin100°) 5(cos 295° + i sin 295°) ÷ 2(cos 25° + i sin 25°) g (cos 250° + i sin 250°) ⋅ h (cos 347° + i sin 347°) ÷ (cos 32° + i sin 32°) (cos 50° + i sin 50°) Match your answers to those in the code below –4 4i I + 4i E D − 2i −3 + 3 i R M − i E −4 + 4i –2.5 i O V Use the code to reveal the name of the French-born mathematician credited with a theorem that uses trigonometric form to find powers of complex numbers and, thus, the n nth roots of a complex number _ _ a b _ _ _ _ _ _ c © Milliken Publishing Company d e f g 40 www.EngineeringBooksPDF.com h MP3510 Name Date Assessment A Score % Trigonometric Functions Shade in the circle of the correct answer y In the diagram, point (x, y) is where the terminal side of angle θ intersects the unit circle If x = − (x, y) θ x –1 –1 , what is one possible value for angle θ ? A 60° B 120° C 135° D 150° If sinθ = A C 11 11 C 12 13 12 B 11 A C 11 © Milliken Publishing Company π π D π 1− k k B 1− k 1− k 1− k k D If cscθ = − and secθ > 0, what is the value of cotθ ? A B –1 12 C 12 13 D − 2 10 If tanθ > and sinθ < 0, which is an expression for cosθ in terms of cotθ ? For which value of θ is this expression undefined? sinθ + cos θ A 120° B 135° C 270° B 5 D E If θ is a positive acute angle and cosθ = k, which is an expression for tanθ in terms of k ? ⎛ 5⎞ What is the value tan ⎜ Arc cos ⎟ ? 13 ⎠ ⎝ A D AE D − C DE C B − B BD C and cosθ = − , what is the value of tanθ ? x θ A A BC A D IV 11 D B ⎛ 1⎞ What is the value of sin−1 ⎜ ⎟ ? ⎝ 2⎠ If sinθ < and secθ > 0, in which quadrant does angle θ terminate? A I B II C III y This diagram shows a unit circle Which line segment has a length equivalent to tanθ ? A cot θ + cot θ C + cot θ B − cot θ + cot θ D − + cot θ D 300° 41 www.EngineeringBooksPDF.com MP3510 Assessment B Name Date Score % Graphs of Trigonometric Functions Shade in the circle of the correct answer What is the period of y = –3 sin 2x ? Which is an equation for this graph? π A B C π D 2π y What is the horizontal shift for the function y = –4 sin (2πx + π)? A π B − C D − –2π π B C y π x 2π A y = sin x B y = sin 2x 1 C y = sin x D y = sin x 2 Which function does NOT have an amplitude of 2? A y = sin x B y = cos x C y = tan x D y = cos 2x B C − − π 3π π 3π , , D −2π , Which statement is NOT true about the function y = + sin (2x – 90°)? A The amplitude is B The period is 180° C The horizontal shift is 45° D The vertical shift is ⎧ y = −3 cos x ⎨ ⎩ y = sin 2x 3π , ⎛ π⎞ y = + sin ⎜ 2x + ⎟ 2⎠ ⎝ ⎛ π⎞ y = + cos ⎜ 2x − ⎟ 2⎠ ⎝ For the interval π ≤ x ≤ π, how many ⎧ y = cos 2x ⎨ values of x satisfy ⎩ y = − sin x this system? A B C D For the interval –2π ≤ x ≤ 2π, which values of x satisfy this system? x What is the range of y = cos 2x ? A –4 ≤ y ≤ B –2 ≤ y ≤ C ≤ x ≤ 2π D –2π ≤ x ≤ 2π –1 , 2π ⎛ ⎞ D y = + cos 2x + π ⎜⎝ ⎟ 2⎠ π π ⎛ π⎞ A y = + sin ⎜ 2x − ⎟ 2⎠ ⎝ Which is an equation for this graph? A –π − − π , 3π π , , − 10 The amplitude of y = sin x + cos x is 3π π , π , © Milliken Publishing Company 3π , 2π A B 2 C D undefined 42 www.EngineeringBooksPDF.com MP3510 Assessment C Name Date Score % Trigonometric Equations and Identities Shade in the circle of the correct answer In the interval ≤ θ < 2π, what is the solution set for this equation? cosθ sinθ + cosθ = sinθ + 1 Which is equivalent to this expression? sin 80° cos 70° + cos 80° sin70° A sin 10° C sin 150° B cos 10° D cos 150° Which is equivalent to this expression? (1 + cosθ)(1 – cosθ) B sin θ D sinθ ⎧ 3π ⎫ C ⎨ ⎬ ⎩2⎭ ⎧π D ⎨ , ⎩2 A C ⎧ C ⎨π , ⎩ 3π 2 ⎫ ⎬ ⎭ ⎧ B ⎨0, π , ⎩ ⎫ ⎬ ⎭ D {0, π} © Milliken Publishing Company B ⎫ ⎬ ⎭ 3π θ ? 5 D In the interval ≤ θ < 2π, what is the solution set for this equation? − cos θ − cos θ = cos2 θ =1 + sinθ 3π ⎫ ⎬ ⎭ , what is the value of cos In the interval ≤ θ < 2π, what is the solution set for this equation? ⎧ A ⎨0, ⎩ 3π 3π B 15 D 60 cos θ = In the interval ≤ θ < 2π , what is the solution set for this equation? |2 sinθ + 5| – = A {–1, 1} ⎧ D ⎨0, ⎩ If θ is a positive acute angle, and B sinθ D secθ ⎧π ⎫ B ⎨ ⎬ ⎩2⎭ ⎧ 3π ⎫ C ⎨ ⎬ ⎩2⎭ A C 30 Which is equivalent to this expression? cos θ sin 2θ A cosθ C cscθ B {0} If sin (2θ + 20)° = cos 40°, what is the value of θ ? A –sin θ C –2 sinθ A ϕ A {–2, 0} B {0} ⎧π ⎫ C ⎨ ⎬ ⎩2⎭ ⎫ ⎧π D ⎨ , π⎬ ⎭ ⎩2 10 In the interval 0° ≤ θ < 360°, what is the solution set for this equation? ⎫ ⎬ ⎭ sin θ = A {30°, 150°} C {60°, 120°} 43 www.EngineeringBooksPDF.com B {150°, 210°} D {60°, 300°} MP3510 Assessment D Name Date Solving Triangles Score % Shade in the circle of the correct answer In ᭝ABC, m∠A = 65°, m∠B = 70°, and side b = in Which is closest to the area of ᭝ABC ? Which condition does NOT prove that two triangles are congruent? A AAS ഡ AAS C ASA ഡ ASA B SSA ഡ SSA D SAS ഡ SAS A in C 16 in In ᭝ABC , a = 17 cm, b = 10 cm, and m∠A = 110° Which equation can be used to find the measure of ∠B ? A sin B = C sin B = 10 B sin B = 17 17 D sin B = 10 Two forces, one of 35 pounds and the other of 70 pounds, act on a body at an angle of 40° What is the magnitude of the resultant force? 10 sin 70° 17 A 49 lb C 89 lb 10 sin 20° 17 B 12 in C 24 in D 24 in A C A 1,202 yd C 811 yd B 81° D 119° , and b = ? A It is an isosceles triangle B It is a right triangle C It is not unique D It cannot be constructed © Milliken Publishing Company B 998 yd D 673 yd 10 A tower, A B, is on a slope inclined at an angle of 15° to the horizontal At a A point C, 80 meters down the slope from the foot of the tower, the tower subtends an angle of 40° B What is the height 40° of the tower? C 80 m Which statement best describes a ᭝ABC that can be constructed if m∠A = 30°, a = B D Boat A is 900 yd directly north of boat B, both sighting a lighthouse at point C The bearing of the lighthouse from A is S 60° E and N 46° E from B What is the distance from A to C ? To the nearest degree, what is the measure of the largest angle of a triangle with sides of length 10, 12, and 18 feet? A 71° C 109° B 69 lb D 99 lb If m∠A = 50°, side a = units, and side b = 10 units, what is the maximum number of distinct triangles that can be constructed? The sides of a parallelogram are inches and inches, and the included angle measures 150° What is the area of the parallelogram? A 12 in B 12 in D 18 in 15° A 90 m C 119 m 44 www.EngineeringBooksPDF.com B 109 m D 129 m MP3510 Answers Page π radians revolution 12 revolution 7π radians 700 grads 6 radians π grad 600 Page 3π cm 2 44.9 in 45 in 57.3° → → → → → Page Q I, Q II, Q IV, sinθ > 0, cotθ > cotθ > 0, cscθ > c o y p o m o c θ is in Q I θ is in Q I θ is in Q I θ is in Q I cscθ > 0, cotθ < cosθ < 0, cscθ > tanθ < 0, cscθ > θ is in Q II θ is in Q II θ is in Q II θ is in Q II cscθ < 0, tanθ > cotθ > 0, cscθ < cotθ > 0, cosθ < sinθ < 0, cotθ > θ is in Q III θ is in Q III θ is in Q III θ is in Q III secθ > 0, sinθ < cscθ < 0, secθ > sinθ < 0, cosθ > tanθ < 0, secθ > θ is in Q IV θ is in Q IV θ is in Q IV θ is in Q IV t a s s e m e n c c e n n n 10 e u 11 s e e c o o s t a r o n t a n g c a n t e y c o n t a s Page a PA b TB c OA d OA e P´A f T´B g RQ h OT n g 12 i n → S → Y → M → M → E → T → R → Y → T → R e = 122.1 f = 20 → O → N g = 63.8 h = 154.2 → O → M − − 17 15 17 15 –100 40 csc 90° 40 cos 270° 20 15 tan 90° undefined 50 – sec 0° 49 SYMMETRY © Milliken Publishing Company –100(cos 180°) 100 – sin 360° 100 100 tan 0° 100 40 + cot 180° 40(cos 0°) (sin 270°) – 40 undefined 40 –41 20 + tan 180° 20(csc 180°) –20(sin 90°) 20 undefined –15(csc 270°) (sin 180°) + 15 15 15 –50(cos 180°) 50 − 1+ 2−3 2+ 2− 2+ 2− 2+3 1− The magic sum is Page 10 Paths will vary; sample path shown (sin 0°) – 100 e n → A → S c = 121.4 d = 112.1 ASTRONOMY − Page 11 t Page a = 18 b = 24 − − a c − 2 o l cosθ > 0, tanθ > sinθ > 0, tanθ < c t s i h secθ > 0, sinθ > 17 − 15 − how wish could calculate pi Q II Q III Q II Sample answer: how I wish I could calculate pi 3.141592 Page Page Q IV, Q IV, Q III, sin 360° 50 2+ 2 − −1 2 +1 2 − +1 2 2 + −1 2 −1 2 + +1 2− ⎛ 2⎞ The magic sum is ⎜ ⎟ ⎝ ⎠ In a 3-by-3 magic cell block, the magic sum is times the middle cell –20 csc 90° 15 1/15 50 + cot 270° 45 www.EngineeringBooksPDF.com 50 MP3510 Page 12 Sample answer: − –2 2 –1 cos 315° cos π sin sec (– 330°) Page 13 Key for line → c → a → e → h 2 Page 20 (a) M (l) T (c) O (i) O (h) O → P → T → O → L → M Page 22 true, true true, true, true, true true, false true, false, false, false false, true true, false, true, false false, false false, false, true, true → Y connections: → b → f → d → g GOOD WORK! d connections: → b → c e → d 10 → e 11 → a c a Page 14 1 [ ] ( ) b Page 23 y [ ] [ ] ( ) Sample answer pattern: The number in each bracket set is the sum of the numbers in the sets of parentheses to which they are connected Page 15 135°, 225° 30°, 210° 120°, 240° 30°, 330° 30°, 150° π 4π , 3 ϕ π 7π 3π 11π , , , 6 0, π π 5π 10 , 4 © Milliken Publishing Company 2 y 2 1 –1 2π x π –1 –2 –2 –3 –3 3 x=π y π 5π , π, , 2π 3 x = 0, π 3π x= , 2 ( ) E E R L G → E PTOLEMY Page 18 Key for line → a → c → e → b → d → a (j) (f) (g) (b) (e) METEOROLOGY sin 225° sin (– 110°) csc (– 240°) 3 − –1 csc 60° 3π cos 230° – cos 50° 3 tan 150° − sin (– 270°) – sin 110° sec 240° sin 210° Page 16 a 3π b 90°, 270° ⎛ 1⎞ c sin−1 ⎜ ⎟ ⎝ 3⎠ π 5π 7π 11π d , , , 6 6 e 60°, 120°, 240°, 300° f Arc tan g ϕ 4 π π x= y 2π x 3π π 5π 3π , , 12 12 1 –1 π 2π x π π –1 x –2 –2 –3 π 5π π 3π 2 2 2π π 5π 3π 12 12 46 www.EngineeringBooksPDF.com π π π π 5π 2π MP3510 Page 24 Page 25 Key for line → c → e → g → b y π –1 2π x Page 28 connections: → f → a → h → d a b − − π 5π c , π, 3 π 3π d 0, , 2 e – cosθ f cosθ g sinθ h – sinθ –2 FABULOUS! –3 x sin x cos x sin x + cos x 0 1 π 1 π –1 –1 3π –1 –1 2π 1 Page 26 a → H b → I c → P d → A e → R f → C g → H h → U i → S y –1 2π x π → C 2+ x sin x 0 0 π –1 π 0 3π 2π –1 –2 0 π –1 2π x –2 –3 x sin x sin x 0 1 0 –1 0 π π 3π 2π 5n i 5,400 m 5,500 m • • © Milliken Publishing Company 114° n b in e i e e g n h t c 10o e l n e r f o u 11r i g h t s Page 30 ASA; b = 7.4 m SSA; m∠ B = no solution AAS; shortest side = 40 cm SSA; m∠ Q = 16° SAS; distance = 438 ft Note: In general, the Law of Cosines is used for cases of SAS However, in this case, the triangle is isosceles and, so, the Law of Sines applies SSA; m∠ B = 42° or 138° 316 ft 15.6 in • • s 4e w o 9z 235 m • t u n s d you’ve c done a the job well • f f s 7t 10 tanθ 236 m n 3o − 119 169 240 161 π , π , 5π , 3π 6 π , 2π , 4π , 3π 3 π , 5π , 3π 6 π 5π 7π 11π , , , 6 6 sin θ cos θ y U L U S i Page 27 → → → → HIPPARCHUS ⎛ π⎞ sin x − cos ⎜ x + ⎟ 2⎠ ⎝ → C Page 29 –3 ⎛ π⎞ cos ⎜ x + ⎟ 2⎠ ⎝ → L CALCULUS –2 → A • • 100° 47 www.EngineeringBooksPDF.com • 80° • 66° Page 31 c = 15.6 in m∠ R = 100° 235 m in m∠ ABC = 114° about 5,400 m MP3510 c In  PQR :  cot α = x + y , so  x + y = h cot α    (equation 1) Z H 2 A M sinθ cos θ sin 90 cos θ + cos 90° sinθ PR = sinθ cos θ 1(cos θ ) + 0(sinθ ) Page 34 a 104 mi b 13 mi c 38 mi d 43 mi e 35 mi f N 55° E g 61 mi ° → A → Z → I → M → U → T → H Q 2θ (90° + θ) θ R (90° – θ) S ABSTRACT AZIMUTH Page 35 Sample answers: a In  PQS :  sinθ = QS = PQ SR In  QRS :  sinθ = QS SR                    sinθ = sinθ                   ∴ SR = sin2 θ b QS = QS RQ RQ so , PQ = PQ tanα TR TR  In  RST :  sin β = = SR PQRT is a rectangle, so TR = PQ In  PQR :  tanα = RQ tanα                                     ∴ RQ = tanα sin β                    sin β = TR = PQ = © Milliken Publishing Company Page 36 Sample answer: The graph seems to be a circle with center at (2, 90˚) and radius = I could use a compass to verify this conjecture Page 38 r = θ = 60° r = cos2θ r = – cosθ r = 16 cos2θ r = – sinθ r = sin3θ r = cos θ + sinθ cos 2θ r = cos θ LEMNISCATE → E → M → N → I → S → C → A → T → E 48 www.EngineeringBooksPDF.com 2 cis i (r, θ) cis 120° 3 − + cis 30° 5π +i r cis θ h −i ) 11π cos θ = r = x2 + y2 Page 40 a + 4i b 4i c −3 + 3 i ( cis 4 π 4 i r sinθ = y i + cis 135° 3 i − 5π −5 2 + 2i r cis θ cis d e f g P − cis 300° cis 315° r(cosθ + i sinθ) PR = sinθ cos θ PR = cos θ ∴ PR = sinθ 3 – 3i sin 2θ sin(90° + θ ) 5 PR = 3 cis 120° − PR = sin(90° + θ ) sin 2θ E i d (alternate) Sample answer: T                                         ∴PR R = sinθ  In  PQR : 4−4 i Page 33 31° 19 lb 47 lb 101° 41° 50 lb − + x = r cosθ cis HERON 8i 2 d  In  PRS :  cos θ = PS PR PS PS In  PQS :   sin 2θ = = PQ                                             PS = sin 2θ                                            PS = sinθ cos θ ⎛ PS ⎞                                            PS = sinθ ⎜ ⎟ ⎝ PR ⎠ y = r sinθ E R O N x +y =r x + yi ⎛ π π⎞ ⎜ cos + i sin ⎟ 2⎠ ⎝ y ,  y = h cot β    (equation 2) h equation 1 - equation 2 :  x + y − y = h cot α − h cot β                                                  ∴ x = h (cot α − cot β ) In  QRS :   cot β = (x, y) → → → → 72 106 1,900 98 b c d e Page 39 Sample answer: h → H a 96 cis 0° Page 32 x r → D → E → M → O → I → V − 2i → R −4 + 4i –4 –2.5i − → E i DE MOIVRE Page 41 B D D B A 10 C C D B B Page 42 1.C 2.D 3.D 4.C 5.C 10 C A D A B Page 43 C B C B B 10 D B B B A Page 44 B B C C B 10 C D A D A MP3510 .. .Trigonometry Reproducibles MP3510 Trigonometry Author: Marilyn Occhiogrosso Editor: Howard Brenner Project Director:... hypotenuse _ A = 12 product of two reciprocals adjacent opposite www.EngineeringBooksPDF.com MP3510 Applying Trigonometry in Right Triangles Name _ Example C Use the information... Double-Angle Identities 27 Half-Angle Identities 28 Applying Trigonometry in Right Triangles Redefining the Trigonometric Functions