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All Textbook Solutions for Trigonometry (MindTap Course List)
89PS90PS91PS92PSIf z is a complex number, show that the product of z and its conjugate is a real number.If z is a complex number, show that the sum of z and its conjugate is a real number.Find sin and cos if the given point lies on the terminal side of . (3,4)96PS97PS98PS99PSSolve triangle ABC given the following information. B=24.2,C=63.8, and b=5.92inches101PS102PS103PS104PS105PS106PS1PS2PS3PS4PS5PS6PS7PS8PS9PS10PS11PS12PS13PS14PS15PS16PS17PS18PS19PS20PS21PS22PS23PS24PS25PS26PS27PS28PSWrite each complex number in standard form. 1 cis 210Write each complex number in standard form. 1 cis 24031PS32PS33PS34PS35PS36PS37PS38PS39PS40PSWrite each complex number in trigonometric form, once using degrees and once using radians. In each case, begin by sketching the graph to help find the argument . 1+i42PS43PS44PS45PS46PS47PS48PS49PS50PS51PS52PS53PS54PS55PS56PS57PS58PS59PS60PS61PS62PS63PS64PS65PS66PS67PS68PSShow that 2 cis 30 and 2 cis (–30) are conjugates.70PS71PS72PS73PS74PS75PS76PSUse Eulers Formula along with the relationship ei(x+y)=eixeiy to derive the sum formulas for both sine and cosine.78PS79PS80PS81PS82PS83PS84PS85PS86PS87PSIn triangle ABC, A=45.6 and b=567 inches. Find B for each given value of a. (You may get one or two values of B. or you may find that no triangle fits the given description.) a=789 inches89PS90PS91PS92PS1PS2PS3PS4PSMultiply. Leave all answers in trigonometric form. 5(cos15+isin15)2(cos25+isin25)6PSMultiply. Leave all answers in trigonometric form. 9(cos115+isin115)4(cos51+isin51)Multiply. Leave all answers in trigonometric form. 7(cos110+isin110)8(cos201+isin201)9PS10PS11PS12PS13PS14PS15PS16PS17PS18PS19PS20PS21PS22PS23PS24PS25PS26PSUse de Moivres Theorem to find each of the following. Write your answer in standard form. (cis12)1028PS29PS30PS31PS32PSUse de Moivres Theorem to find each of the following. Write your answer in standard form. (3+i)434PS35PS36PS37PS38PSUse de Moivres Theorem to find the reciprocal of each number below. 1+i40PS41PS42PS43PS44PS45PS46PSDivide. Leave your answers in trigonometric form. 4cis28cis648PS49PS50PS51PS52PS53PS54PS55PS56PS57PS58PS59PS60PS61PS62PS63PS64PS65PS66PS67PS68PS69PS70PS71PS72PS73PS74PS75PS76PSMultiply 5(cos10+isin10)4(cos15+isin15). a. 9(cos150+isin150) b. cos100+isin100 c. 20(cos150+isin150) d. 20(cos25+isin25)Divide 8(cos120+isin120)2(cos40+isin40). a. 4(cos30+isin30) b. 4(cos80+isin80) c. 4(cos160+isin160) d. 6(cos30+sin30)79PSSimplify (1i)4(3i)2(1+i3)5 ,_ by first writing each complex number in trigonometric form. Convert your answer back to standard form. a. 34+14i b. 12+32i c. 2222i d. 1434iFor Questions 1 through 4, fill in the blank with an appropriate word, number, or expression. Every complex number has ____ distinct nth roots.For Questions 1 through 4, fill in the blank with an appropriate word, number, or expression. To find the principle nth root of a complex number, find the nth root of the _________ and divide the ___________ by _____. To find the remaining nth roots, add ________ to the argument repeatedly until you have them all.For Questions 1 through 4, fill in the blank with an appropriate word, number, or expression. The 5th roots of a complex number would have arguments that differ from each other by ________ degrees.For Questions 1 through 4, fill in the blank with an appropriate word, number, or expression. When graphed, all of the nth roots of a complex number will be evenly distributed around a ______ of _____ r1n.Find the two square roots for each of the following complex numbers. Leave your answers in trigonometric form. In each case, graph the two roots. 4(cos30+isin30)6PSFind the two square roots for each of the following complex numbers. Leave your answers in trigonometric form. In each case, graph the two roots. 25(cos210+isin210)8PSFind the two square roots for each of the following complex numbers. Leave your answers in trigonometric form. In each case, graph the two roots. 49 cisFind the two square roots for each of the following complex numbers. Leave your answers in trigonometric form. In each case, graph the two roots. 81 cis 5611PS12PSFind the two square roots for each of the following complex numbers. Write your answers in standard form. 4i14PS15PS16PS17PSFind the two square roots for each of the following complex numbers. Write your answers in standard form. 1i3Find the three cube roots for each of the following complex numbers. Leave your answers in trigonometric form. 8(cos210+isin210)20PS21PS22PSFind the three cube roots for each of the following complex numbers. Leave your answers in trigonometric form. –27Find the three cube roots for each of the following complex numbers. Leave your answers in trigonometric form. 825PS26PS27PS28PSSolve each equation. x4+81=030PSFind the four fourth roots of z=16(cos23+isin23). Write each root in standard form.Find the four fourth roots of z=cos43+isin43. Leave your answers in trigonometric form.Find the five fifth roots of z=105cis15. Write each root in trigonometric form and then give a decimal approximation, accurate to the nearest hundredth, for each one.Find the five fifth roots of z=1010cis75. Write each root in trigonometric form and then give a decimal approximation, accurate to the nearest hundredth, for each one.Find the six sixth roots of z=1. Leave your answers in trigonometric form. Graph all six roots on the same coordinate system.36PSSolve each of the following equations. Leave your solutions in trigonometric form. x42x2+4=038PS39PS40PSRecall from the introduction to Section 8.2 that Jerome Cardans solutions to the equation x3=15x+4 could be written as x=2+11i3+211i3 Lets assume that the two cube roots are complex conjugates. If they are, then we can simplify our work by noticing that x=2+11i3+211i3=a+bi+abi=2a which means that we simply double the real part of each cube root of 2+11i to find the solutions to x3=15x+4. Now, to end our work with Cardan, find the three cube roots of 2+11i. Then, noting the discussion above, use the three cube roots to solve the equation x3=15x+4. Write your answers accurate to the nearest thousandth.42PS43PS44PS45PS46PS47PS48PS49PS50PSFind the two square roots of 36i. a. 32+3i2,323i2 b. 32+3i2,323i2 c. 6i,6i d. 6,652PS53PS54PSFor Questions 1 through 6, fill in the blank with an appropriate word, expression, or equation. The polar coordinate system consists of a point. called the _____. and a ray extending out from it. called the _______ _____For Questions 1 through 6, fill in the blank with an appropriate word, expression, or equation. In polar coordinates. r is the ________ ________ on the terminal side of an angle whose vertex is at the _____ and whose initial side lies along the ________ ______3PS4PS5PS6PSFor Questions 7 and 8, determine if the statement is true or false. In rectangular coordinates, each point is represented by a unique ordered pair.For Questions 7 and 8, determine if the statement is true or false. In polar coordinates, each point is represented by a unique ordered pair.9PS10PS11PSGraph each ordered pair on a polar coordinate system. (4,135)13PS14PSGraph each ordered pair on a polar coordinate system. (3,45)Graph each ordered pair on a polar coordinate system. (4,60)17PS18PS19PS20PS21PSFor each ordered pair, give three other ordered pairs with between –360 and 360 that name the same point. (1,120)For each ordered pair, give three other ordered pairs with between –360 and 360 that name the same point. (5,135)24PS25PS26PS27PS28PS29PS30PS31PS32PSConvert to rectangular coordinates. Use exact values. (2,135)34PS35PS36PS37PS38PS39PS40PS41PS42PS43PS44PS45PS46PS47PS48PS49PS50PS51PS52PSConvert to polar coordinates. Use a calculator to find to the nearest tenth of a degree. Keep r positive and between 0 and 360. (2,3)54PS55PS56PS