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Trigonometry (MindTap Course List)

8th Edition
Charles P. McKeague + 1 other
Publisher: Cengage Learning
ISBN: 9781305652224
BuyFindarrow_forward

Trigonometry (MindTap Course List)

8th Edition
Charles P. McKeague + 1 other
Publisher: Cengage Learning
ISBN: 9781305652224
Chapter 8.3, Problem 2PS
Textbook Problem
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For Questions 1 through 4, fill in the blank with an appropriate word or symbol.

To divide two complex numbers in trigonometric form, ________ their moduli and _____ their arguments.

To determine

To fill:

Appropriate word in the blank so that the given sentence makes sense.

Explanation of Solution

Given information:

To divide two complex numbers in trigonometric form, ______their moduli and ______ their arguments.

If z1=r1(cosθ1+isinθ1)=r1cisθ1 and z2=r2(cosθ2+isinθ2)=r2cisθ2 are two complex numbers in trigonometric form, then their product, z1/z2, is

z1z2=r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)        =r1r2[(cos(θ1θ2)+isin(θ1θ2))]         =r1r2cis(θ1θ2)

In words: To divide two complex numbers in trigonometric form, dividing the moduli and subtracting the arguments.

Proof:

As was the case with division of complex numbers in standard form, the major step in this proof is multiplying the numerator and denominator of our quotient by the conjugate of the denominator.

z1z2=r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)       =r1(cosθ1+isinθ1)(cosθ

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Chapter 8 Solutions

Trigonometry (MindTap Course List)
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Ch. 8.1 - Match each power of i with its corresponding value...Ch. 8.1 - For Questions 12 through 14, determine if the...Ch. 8.1 - For Questions 12 through 14, determine if the...Ch. 8.1 - For Questions 12 through 14, determine if the...Ch. 8.1 - Write each expression in terms of i. 16Ch. 8.1 - Write each expression in terms of i. 64Ch. 8.1 - Write each expression in terms of i. 121Ch. 8.1 - Write each expression in terms of i. 225Ch. 8.1 - Write each expression in terms of i. 18Ch. 8.1 - Write each expression in terms of i. 48Ch. 8.1 - Write each expression in terms of i. 8Ch. 8.1 - Write each expression in terms of i. 21Ch. 8.1 - Write in terms of i and then simplify. 49Ch. 8.1 - Write in terms of i and then simplify. 2536Ch. 8.1 - Write in terms of i and then simplify. 19Ch. 8.1 - Write in terms of i and then simplify. 1649Ch. 8.1 - Find x and y so that each of the following...Ch. 8.1 - Find x and y so that each of the following...Ch. 8.1 - Find x and y so that each of the following...Ch. 8.1 - Find x and y so that each of the following...Ch. 8.1 - Find all x and y (0x,y2) so that each of the...Ch. 8.1 - Find all x and y (0x,y2) so that each of the...Ch. 8.1 - Find all x and y (0x,y2) so that each of the...Ch. 8.1 - Find all x and y (0x,y2) so that each of the...Ch. 8.1 - Combine the following complex numbers....Ch. 8.1 - Combine the following complex numbers. 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Then...Ch. 8.3 - Convert all complex numbers to trigonometric form...Ch. 8.3 - Convert all complex numbers to trigonometric form...Ch. 8.3 - Convert all complex numbers to trigonometric form...Ch. 8.3 - Convert all complex numbers to trigonometric form...Ch. 8.3 - Show that x=2(cos60+isin60) is a solution to the...Ch. 8.3 - Show that x=2(cos300+isin300) is a solution to the...Ch. 8.3 - Show that w=2(cos15+isin15) is a fourth root of...Ch. 8.3 - Show that x=12+(32)i is a cube root of –1.Ch. 8.3 - In the Problem Set for Section 8.2, introduced...Ch. 8.3 - In the Problem Set for Section 8.2, introduced...Ch. 8.3 - In the Problem Set for Section 8.2, introduced...Ch. 8.3 - In the Problem Set for Section 8.2, introduced...Ch. 8.3 - If cosA=13 and A is between 90 and 180, find cos...Ch. 8.3 - If cosA=13 and A is between 90 and 180, find sin...Ch. 8.3 - If cosA=13 and A is between 90 and 180, find sinA2Ch. 8.3 - If cosA=13 and A is between 90 and 180, find cosA2Ch. 8.3 - Distance and Bearing A crew member on a fishing...Ch. 8.3 - Length A tent pole is held in place by two guy...Ch. 8.3 - True Course and Speed A plane is flying with an...Ch. 8.3 - Geometry If a parallelogram has sides of 33...Ch. 8.3 - Multiply 5(cos10+isin10)4(cos15+isin15). a....Ch. 8.3 - Divide 8(cos120+isin120)2(cos40+isin40). a....Ch. 8.3 - Find (4cis6)3. a. 12cis2 b. 64cis3216 c. 64cis2 d....Ch. 8.3 - Simplify (1i)4(3i)2(1+i3)5 ,_ by first writing...Ch. 8.4 - For Questions 1 through 4, fill in the blank with...Ch. 8.4 - For Questions 1 through 4, fill in the blank with...Ch. 8.4 - For Questions 1 through 4, fill in the blank with...Ch. 8.4 - For Questions 1 through 4, fill in the blank with...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the two square roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Find the three cube roots for each of the...Ch. 8.4 - Solve each equation. x3+8=0Ch. 8.4 - Solve each equation. x327=0Ch. 8.4 - Solve each equation. x4+81=0Ch. 8.4 - Solve each equation. x416=0Ch. 8.4 - Find the four fourth roots of z=16(cos23+isin23)....Ch. 8.4 - Find the four fourth roots of z=cos43+isin43....Ch. 8.4 - Find the five fifth roots of z=105cis15. 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Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Convert to rectangular coordinates. Use exact...Ch. 8.5 - Use your graphing calculator to convert to...Ch. 8.5 - Use your graphing calculator to convert to...Ch. 8.5 - Use your graphing calculator to convert to...Ch. 8.5 - Use your graphing calculator to convert to...Ch. 8.5 - Convert to polar coordinates with r0 and 0360....Ch. 8.5 - Convert to polar coordinates with r0 and 0360....Ch. 8.5 - Convert to polar coordinates with r0 and 0360....Ch. 8.5 - Convert to polar coordinates with r0 and 0360....Ch. 8.5 - Convert to polar coordinates with r0 and 02. (2,0)Ch. 8.5 - Convert to polar coordinates with r0 and 02. (2,0)Ch. 8.5 - Convert to polar coordinates with r0 and 02. (3,1)Ch. 8.5 - Convert to polar coordinates with r0 and 02. (1,3)Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Convert to polar coordinates. Use a calculator to...Ch. 8.5 - Use your graphing calculator to convert to polar...Ch. 8.5 - Use your graphing calculator to convert to polar...Ch. 8.5 - Use your graphing calculator to convert to polar...Ch. 8.5 - Use your graphing calculator to convert to polar...Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation with rectangular coordinates....Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Write each equation in polar coordinates. Then...Ch. 8.5 - Graph one complete cycle of each equation. y=6sinxCh. 8.5 - Graph one complete cycle of each equation. y=6cosxCh. 8.5 - Graph one complete cycle of each equation....Ch. 8.5 - Graph one complete cycle of each equation....Ch. 8.5 - Graph one complete cycle of each equation....Ch. 8.5 - Graph one complete cycle of each equation....Ch. 8.5 - Which of the following ordered pairs is not a...Ch. 8.5 - Convert (2,56) to rectangular coordinates. a....Ch. 8.5 - Write r=6(cossin) in rectangular coordinates. a....Ch. 8.5 - Write x2+xy+y2=1 in polar coordinates, and isolate...Ch. 8.6 - For Questions 1 through 3, fill in the blank with...Ch. 8.6 - For Questions 1 through 3, fill in the blank with...Ch. 8.6 - For Questions 1 through 3, fill in the blank with...Ch. 8.6 - Match each equation with its appropriate graph....Ch. 8.6 - Sketch the graph of each equation by making a...Ch. 8.6 - Sketch the graph of each equation by making a...Ch. 8.6 - Sketch the graph of each equation by making a...Ch. 8.6 - Sketch the graph of each equation by making a...Ch. 8.6 - Use your graphing calculator in polar mode to...Ch. 8.6 - Use your graphing calculator in polar mode to...Ch. 8.6 - Use your graphing calculator in polar mode to...Ch. 8.6 - Use your graphing calculator in polar mode to...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Determine whether the graph of the given equation...Ch. 8.6 - Graph each equation. r=3Ch. 8.6 - Graph each equation. x=2Ch. 8.6 - Graph each equation. =34Ch. 8.6 - Graph each equation. =4Ch. 8.6 - Graph each equation. r=3sinCh. 8.6 - Graph each equation. r=3cosCh. 8.6 - Graph each equation. r=4+2sinCh. 8.6 - Graph each equation. r=4+2cosCh. 8.6 - Graph each equation. r=2+4cosCh. 8.6 - Graph each equation. r=2+4sinCh. 8.6 - Graph each equation. r=2+2sinCh. 8.6 - Graph each equation. r=2+2cosCh. 8.6 - Graph each equation. r2=4cos2Ch. 8.6 - Graph each equation. r2=9sin2Ch. 8.6 - Graph each equation. r=2sin2Ch. 8.6 - Graph each equation. r=2cos2Ch. 8.6 - Graph each equation. r=4cos3Ch. 8.6 - Graph each equation. r=4sin3Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Graph each equation using your graphing calculator...Ch. 8.6 - Convert each equation to polar coordinates and...Ch. 8.6 - Convert each equation to polar coordinates and...Ch. 8.6 - Convert each equation to polar coordinates and...Ch. 8.6 - Convert each equation to polar coordinates and...Ch. 8.6 - Convert each equation to polar coordinates and...Ch. 8.6 - Convert each equation to polar coordinates and...Ch. 8.6 - Change each equation to rectangular coordinates...Ch. 8.6 - Change each equation to rectangular coordinates...Ch. 8.6 - Change each equation to rectangular coordinates...Ch. 8.6 - Change each equation to rectangular coordinates...Ch. 8.6 - Change each equation to rectangular coordinates...Ch. 8.6 - Change each equation to rectangular coordinates...Ch. 8.6 - Graph r1=2sin and r2=2cos and then name two points...Ch. 8.6 - Graph r1=2+2cos and r2=22cos and name three points...Ch. 8.6 - Graph each equation. y=sinxcosx,0x4Ch. 8.6 - Graph each equation. y=cosxsinx,0x4Ch. 8.6 - Graph each equation. y=x+sinx,0x8Ch. 8.6 - Graph each equation. y=x+cosx,0x8Ch. 8.6 - Graph each equation. y=3sinx+cos2x,0x4Ch. 8.6 - Graph each equation. y=sinx+12cos2x,0x4Ch. 8.6 - Table 4 shows ordered pairs for a polar equation....Ch. 8.6 - Use the rectangular graph of r=f() shown in Figure...Ch. 8.6 - Use your graphing calculator to determine which of...Ch. 8.6 - Which equation has a graph that is a four-leaved...Ch. 8 - Write 12 in terms of i.Ch. 8 - Find x and y so that the equation...Ch. 8 - Simplify (63i)+[ (42i))(3+i) ]..Ch. 8 - Simplify i67.Ch. 8 - Multiply. Leave your answer in standard form....Ch. 8 - Multiply. Leave your answer in standard form....Ch. 8 - Divide 6+5i65i. Write your answer in standard...Ch. 8 - For each of the following complex numbers give:...Ch. 8 - For each of the following complex numbers give:...Ch. 8 - Write each complex number in standard form....Ch. 8 - Write each complex number in standard form. 2cis34Ch. 8 - Write each complex number in trigonometric form....Ch. 8 - Write each complex number in trigonometric form....Ch. 8 - Multiply or divide as indicated. Leave your...Ch. 8 - Multiply or divide as indicated. Leave your...Ch. 8 - Multiply or divide as indicated. Leave your...Ch. 8 - Find the two square roots of z=49(cos50+isin50)....Ch. 8 - Find the four fourth roots of z=2+2i3. Leave your...Ch. 8 - Solve each equation. Write your solutions in...Ch. 8 - Solve each equation. Write your solutions in...Ch. 8 - Convert the point (6,60) to rectangular...Ch. 8 - Convert (3,3) to polar coordinates with r positive...Ch. 8 - Convert the equation r=5cos to rectangular...Ch. 8 - Convert the equation x2+y2=8y to polar...Ch. 8 - Graph each equation. r=4Ch. 8 - Graph each equation. r=4+2cosCh. 8 - Graph each equation. r=sin2Ch. 8 - Graph each equation using your graphing calculator...Ch. 8 - Graph each equation using your graphing calculator...Ch. 8 - Graph each equation using your graphing calculator...Ch. 8 - We can approach the problem as a system of two...Ch. 8 - Verify your results from Question I graphically....Ch. 8 - Use the substitution method to find any points of...Ch. 8 - Verify your results graphically. Sketch the graph...Ch. 8 - Show that the pole r=0 does satisfy each equation...Ch. 8 - Use your graphing calculator to graph both...Ch. 8 - Based on your observations of the graphs, try to...Ch. 8 - As was mentioned in the introduction to this...Ch. 8 - A Bitter Dispute With the publication of Ars...Ch. 8 - If cos=35 with in QIV, find the values of the...Ch. 8 - Use a calculator to find to the nearest tenth of...Ch. 8 - Use a calculator to find . to the nearest tenth...Ch. 8 - Height of a Tree An arborist needs to measure the...Ch. 8 - If triangle ABC is a right triangle with...Ch. 8 - Convert 49 to degree measure.Ch. 8 - If =23 is a central angle that cuts off an arc...Ch. 8 - If a point is rotating with uniform circular...Ch. 8 - Use the graph of y=secx to find all values of x...Ch. 8 - Graph one complete cycle of y=12sin2x.Ch. 8 - Write an equivalent algebraic expression for...Ch. 8 - Prove the equation sec2xcsc2x=sec2x+csc2x is an...Ch. 8 - Find exact values for each of the following: cos15Ch. 8 - Find exact values for each of the following: cot12Ch. 8 - Solve 3cos2+5sin=0 for if 0360. Round your...Ch. 8 - Problems 16 through 18 refer to triangle ABC,...Ch. 8 - Problems 16 through 18 refer to triangle ABC,...Ch. 8 - Problems 16 through 18 refer to triangle ABC,...Ch. 8 - Problems 16 through 18 refer to triangle ABC,...Ch. 8 - For Problems 20 and 21 use the vectors U=5i+12j...Ch. 8 - For Problems 20 and 21 use the vectors U=5i+12j...Ch. 8 - Simplify (7+3i)[ (2+i)(34i) ].Ch. 8 - Simplify i16.Ch. 8 - For the complex number 3+4i , give (a) the...Ch. 8 - Write 2+2i in trigonometric form.Ch. 8 - Simplify [ 2(cos10+isin10) ]5. Give your answer in...Ch. 8 - Find the two square roots of 44i3. Write your...Ch. 8 - Convert the point (4,225) to rectangular...Ch. 8 - Convert the equation x+y=2 to polar coordinates.Ch. 8 - Graph the equation r=2cos2+3sin using your...

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