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All Textbook Solutions for Algebra and Trigonometry (MindTap Course List)

65E 66E 67-72. Value of a Product or Sum Find the value of the product or sum. 2sin52.5sin97.5 67-72. Value of a Product or Sum Find the value of the product or sum. 3cos37.5cos7.5 69E 67-72. Value of a Product or Sum Find the value of the product or sum. sin75+sin15 71E 72E 73-92. Proving Identities Prove the identity. cos25xsin25x=cos10x 74E 75E 76E 77E 78E 73-92. Proving Identities Prove the identity. tan(x2)+cosxtan(x2)=sinx 80E 81E 82E 83E 84E 73-92 Proving Identities Prove the identity. cot2x=1tan2x2tanx 86E 87E 88E 89E 90E 73-92 Proving Identities Prove the identity. sin10xsin9x+sinx=cos5xcos4x 92E 93E 94E 95E 96E 97E 97-100. Sum to product formulas Use a Sum-to-Product Formula to show the following. cos100cos200=sin50 99E 100E 101E 102E 103E 104E 105E 106E 107E 108E 109E Length of a Bisector In triangle ABC see the figure the line segment s bisects angle C. Show that the length of s is given by s=2abcosxa+b Hint: Use the Law of Sines.111E 112E 113E APPLICATIONS Length of a Fold The lower right-hand corner of a long piece of paper 6in, wide is folded over to the left-hand edge as shown. The length L of the fold depends on the angle . Show that L=3sincos2 115E APPLICATIONS Touch-Tone Telephones When a key is pressed on a touch-tone telephone, the keypad generates two pure tones, which combine to produce a sound that uniquely identifies the key. The figure shows the low frequency f1 and the high frequency f2 associated with each key. Pressing a key produces the sound wave y=sin(2f1t)+sin(2f2t) a Find the function that models the sound produced when the 4 key is pressed. b Use a Sum-to-Product Formula to express the sound generated by the 4 key as a product of a sine and a cosine function. c Graph the sound wave generated by the 4 key from t=0 to t=0.006s 117E Because the trigonometry functions are periodic, if a basic trigonometric equation has one solution, it has ________several/infinitely many solutions.The basic equation sinx=2 has_________no/one/infinitely many solutions, whereas the basic equation sinx=0.3 has _______no/one/infinitely many solutions.3E We can find the solutions of sinx=0.3 algebraically. a First we find the solutions in the interval [0,2). We get one such solution by taking sin1 to get x_______. The other solution in this interval is x_______. b We find all solutions by adding multiples of ______ to the solutions in [0,2). The solutions are x_______ and x_______.5E 6E 7E 5-16 Solving Basic Trigonometric Equations Solve the given equation. cos=32 9E 10E 5-16 Solving Basic Trigonometric Equations Solve the given equation. sin=0.45 12E 13E 5-16 Solving Basic Trigonometric Equations Solve the given equation. tan=1 15E 16E 17-24 Solving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. cos=32 18E 19E 17-24 Solving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. sin=32 21E 22E 23E 17-24 Solving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. sin=0.9 25E 26E 27E 25-38 Solving Basic Trigonometric Equations Find all solutions of the given equation. 2cos1=0 29E 25-38 Solving Basic Trigonometric Equations Find all solutions of the given equation. 4cos+1=0 31E 25-38 Solving Basic Trigonometric Equations Find all solutions of the given equation. cot+1=0 33E 34E 35E 36E 25-38 Solving Basic Trigonometric Equations Find all solutions of the given equation. sec22=0 38E 39E 40E 41E 39-56 Solving Basic Trigonometric Equations by Factoring Solve the given equation. 2sin2sin1=0 43E 44E 45E 39-56 Solving Basic Trigonometric Equations by Factoring Solve the given equation. sin2sin2=0 47E 48E 49E 50E 51E 52E 53E 54E 39-56 Solving Basic Trigonometric Equations by Factoring Solve the given equation. 3tansin2tan=0 56E 57E Total Internal Reflection When light passes from a more dense to a less dense medium-from glass to air, for example-the angle of refraction predicted by Snells Law see exercise 57 can be 90 or large. In this case the light beam is actually reflected Bach into the denser medium. This phenomenon, called total internal reflection, is the principal behind optics. Set 2=90 in Snells Law, and solve for 1 to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. Note that the index of refraction from glass to air is reciprocal of the index from air to glass.59E 60E 1.2 We can use identities to help us solve trigonometric equations. Using a Pythagorean identity we that the equation sinx+sin2x+cos2=1 is equivalent to basic equation whose solution are x=.2E 3E 3-16 Solving trigonometric equations by using identities. Solve the given equation. sin2=42cos2 5E 6E SKILLS 3-16 Solving Trigonometric Equations By Using Identities Solve the given equation. 2sin23sin=0 SKILLS 3-16 Solving Trigonometric Equations By Using Identities Solve the given equation. 3sin22sin=0 9E SKILLS 3-16 Solving Trigonometric Equations By Using Identities Solve the given equation. cos2=cos212 SKILLS 3-16 Solving Trigonometric Equations By Using Identities Solve the given equation. 2sin2cos=1 12E SKILLS 3-16 Solving Trigonometric Equations By Using Identities Solve the given equation. sin1=cos 14E 15E SKILLS 3-16 Solving Trigonometric Equations By Using Identities Solve the given equation. 2tan+sec2=4 17E 18E 17-30Solving a Trigonometric Equation Involving Multiple of an Angle. An equation is given a Find all the solution of the equation. b Find the solutions in the interval [0,2). 2cos2+1=0 20E 21E 17-30Solving a Trigonometric Equation Involving Multiple of an Angle. An equation is given a Find all the solution of the equation. b Find the solutions in the interval [0,2). sec42=0 23E 17-30Solving a Trigonometric Equation Involving Multiple of an Angle. An equation is given a Find all the solution of the equation. b Find the solutions in the interval [0,2). tan4+3=0 17-30Solving a Trigonometric Equation Involving Multiple of an Angle. An equation is given a Find all the solution of the equation. b Find the solutions in the interval [0,2). 2sin3+3=0 26E 27E 28E 17-30Solving a Trigonometric Equation Involving Multiple of an Angle. An equation is given a Find all the solution of the equation. b Find the solutions in the interval [0,2). 12sin=cos2 30E 31-34Solving Trigonometric Equations Solve the equations by factoring. 3tan33tan2tan+1=0 31-34Solving Trigonometric Equations Solve the equations by factoring. 4sincos+2sin2cos1=0 33E 34E 35E 36E 35-38 Finding Intersection Points Graphically. a Graph f and g in the given viewing rectangle and find the intersection point graphically. b Find the intersection points of f and g algebraically. Give exact answer. f(x)=tanx,g(x)=3; [2,2]by[10,10]38E 39E 40E 41E 42E 43-52 Using Double- or Half-Angle FormulasUse a Double- angle or Half-Angle Formula to solve the equation in the interval [0,2). sin2+cos=0 44E 45E 43-52 Using Double or Half Angle formulas. Use a double angle or Half Angle formula to solve the equation in the interval. [0,2]. tan+cot=4sin2 47E 48E 43-52 Using Double-or Half-Angle Formulas Use a Double or Half-Angled Formula to solve the equation in the interval [0,2). cos2cos4=0 50E 51E 52E 53E 53-56 Using Sum-to-Product Formulas Solve the equation by first using a Sum-to-Product Formula. cos5cos7=0 53-56 Using Sum-to-Product FormulasSolve the equation by first using a Sum-to-Product Formula. cos4+cos2=cos 56E 57E 58E 59E 60E 57-62 Solving Trigonometric EquationsGraphically Use a graphing device to find the solutions of the equation, rounded to two decimal palces. cosx1+x2=x2 62E 63E 64E 65E 66E 67E Belts and Pulleys A thin belt of length L surrounds two pulleys of radii R and r, as shown in the figure (1). aShow that the angle in rad where the belt crosses itself satisfies the equation +2cot2=LR+r Hint: Express L in terms of R, r, and by adding up the lengths of the curved and straight parts of the belt. bSuppose that R=2.42ft, r=1.21ft, and L=27.78ft. Find by solving the equation in part a graphically. Express your answer both in radians and in degrees. Figure (1)69E What is an identity? What is a trigonometric identity?2CC 3CC 4CC 5CC 6CC 7CC 8CC 9CC 10CC a State the Sum-to-Product Formula for the sum sinx+siny b Express sin5x+sin7x as a product of trigonometric functions.12CC 1CR 2CR 3CR 4CR 5CR 6CR 7CR 8CR 9CR 10CR 11CR 12CR 13CR 14CR 15CR 16CR 17CR 18CR 19CR 20CR 21CR 22CR 23CR 24CR 25CR 26CR 27CR 28CR 29CR 30CR 31CR 32CR 33CR 34CR 35CR 36CR 37CR 38CR 39CR 40CR 41CR 42CR 43CR 44CR 45CR 46CR 47CR 48CR 49CR 50CR 51CR 52CR 53CR 54CR 55CR 56CR 57CR 58CR 59CR 60CR 61CR 62CR 63CR 64CR 65CR 66CR 67CR 68CR 69CR Viewing Angle of a Tower A 380-ft-tall building supports a 40-ft communications tower see the figure. As a driver approaches the building, the viewing angle of the tower changes. a.Express the viewing angle as a function of the distance x between the driver and the building. b.At what distance from the building is the viewing angle as large as possible?1-8 Verify each identity. tansin+cos=sec 2CT 3CT 4CT 5CT 6CT 7CT 8CT Find the exact value of each expression. a sin8cos22+cos8sin22 b sin75 c sin(12)For the angles and in the figures, find cos(+).Write sin3xcos5x as a sum of trigonometric functions.12CT 13CT 14CT 15CT 16CT 17CT 18CT 19CT 20CT 21CT 22CT WAVE ON A CANAL A wave on the surface a long canal us described by the function y(x,t)=5sin(2x2t), x0 (a) Find the function that models the positions of the point x=0 at any time t. (b) Sketch the shape of the wave when t=0,0.4,0.8,1.2 and 1.6. Is this a travelling wave? (c) Find the velocity of the wave.2P 3P 4P 5P 6P Vibrating String When a violin string vibrates, the sound produced results from a combination of standing waves that have evenly placed nodes. The figure (1) illustrates some of the possible standing waves. Lets assume that the string has length . a.For fixed t, the string has the shape of a sine curve y=Asinx. Find the appropriate value of for each of the illustrated standing waves. b.Do you notice a pattern in the values of that you found in part a? What would the next two values of be? Sketch rough graphs of the standing waves associated with these new values of . c.Suppose that for fixed t, each point on the string that is not a node vibrates with frequency 440Hz. Find the value of for which an equation of the form y=Acost would model this motion. d.Combine your answers for parts a and c to find functions of the form y(x,t)=Asinxcost that model each of the standing waves in the figure. Assume that A=1. Figure (1)8P CONCEPTS We can describe the location of a point in the plane using different _______ system. The point P shown in the figure has rectangular coordinates _, _ and polar coordinates _, _ .2E 3E 4E 5E 6E

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