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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Try It

10.1 Non-right Triangles: Law of Sines

1.

α= 98 a=34.6 β= 39 b=22 γ= 43 c=23.8 α= 98 a=34.6 β= 39 b=22 γ= 43 c=23.8

2.

Solution 1

α=80° a=120 β83.2° b=121 γ16.8° c35.2 α=80° a=120 β83.2° b=121 γ16.8° c35.2

Solution 2

α =80° a =120 β 96.8° b =121 γ 3.2° c 6.8 α =80° a =120 β 96.8° b =121 γ 3.2° c 6.8
3.

β5.7°,γ94.3°,c101.3 β5.7°,γ94.3°,c101.3

4.

two

5.

about 8.2squarefeet 8.2squarefeet

6.

161.9 yd.

10.2 Non-right Triangles: Law of Cosines

1.

a14.9,β23.8°,γ126.2°. a14.9,β23.8°,γ126.2°.

2.

α27.7°,β40.5°,γ111.8° α27.7°,β40.5°,γ111.8°

3.

Area = 552 square feet

4.

about 8.15 square feet

10.3 Polar Coordinates

1.
Polar grid with point (2, pi/3) plotted.
2.
Points (2, 9pi/4) and (3, -pi/6) are plotted in the polar grid.
3.

( x,y )=( 1 2 , 3 2 ) ( x,y )=( 1 2 , 3 2 )

4.

r= 3 r= 3

5.

x 2 + y 2 =2y x 2 + y 2 =2yor, in the standard form for a circle, x 2 + ( y1 ) 2 =1 x 2 + ( y1 ) 2 =1

10.4 Polar Coordinates: Graphs

1.

The equation fails the symmetry test with respect to the line θ= π 2 θ= π 2 and with respect to the pole. It passes the polar axis symmetry test.

2.

Tests will reveal symmetry about the polar axis. The zero is ( 0, π 2 ), ( 0, π 2 ), and the maximum value is (3,0). (3,0).

3.
Graph of the limaçon r=3-2cos(theta). Extending to the left.
4.

The graph is a rose curve, n n even

Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4.
5.
Graph of rose curve r=3cos(3theta). Three petals equally spaced from origin.

Rose curve, n n odd

6.

10.5 Polar Form of Complex Numbers

1.
Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).
2.

13

3.

| z |= 50 =5 2 | z |= 50 =5 2

4.

z=3( cos( π 2 )+isin( π 2 ) ) z=3( cos( π 2 )+isin( π 2 ) )

5.

z=2( cos( π 6 )+isin( π 6 ) ) z=2( cos( π 6 )+isin( π 6 ) )

6.

z=2 3 2i z=2 3 2i

7.

z 1 z 2 =4 3 ; z 1 z 2 = 3 2 + 3 2 i z 1 z 2 =4 3 ; z 1 z 2 = 3 2 + 3 2 i

8.

z 0 =2(cos(30°)+isin(30°)) z 0 =2(cos(30°)+isin(30°))

z 1 =2(cos(120°)+isin(120°)) z 1 =2(cos(120°)+isin(120°))

z 2 =2(cos(210°)+isin(210°)) z 2 =2(cos(210°)+isin(210°))

z 3 =2(cos(300°)+isin(300°)) z 3 =2(cos(300°)+isin(300°))

10.6 Parametric Equations

1.

t t x( t ) x( t ) y( t ) y( t )
1 1 4 4 2 2
0 0 3 3 4 4
1 1 2 2 6 6
2 2 1 1 8 8
2.

x(t)= t 3 2t y(t)=t x(t)= t 3 2t y(t)=t

3.

y=5 1 2 x3 y=5 1 2 x3

4.

y=ln x y=ln x

5.

x 2 4 + y 2 9 =1 x 2 4 + y 2 9 =1

6.

y= x 2 y= x 2

10.7 Parametric Equations: Graphs

1.
Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.
2.
Graph of the given equations - a horizontal ellipse.
3.

The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.

Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.

10.8 Vectors

1.
A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.
2.

3u= 15,12 3u= 15,12

3.

u=8i11j u=8i11j

4.

v= 34 cos(59°)i+ 34 sin(59°)j v= 34 cos(59°)i+ 34 sin(59°)j

Magnitude = 34 34

θ= tan 1 ( 5 3 )=59.04° θ= tan 1 ( 5 3 )=59.04°

10.1 Section Exercises

1.

The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.

3.

When the known values are the side opposite the missing angle and another side and its opposite angle.

5.

A triangle with two given sides and a non-included angle.

7.

 β=72°,a12.0,b19.9  β=72°,a12.0,b19.9

9.

 γ=20°,b4.5,c1.6  γ=20°,b4.5,c1.6

11.

b3.78 b3.78

13.

c13.70 c13.70

15.

one triangle, α50.3°,β16.7°,a26.7 α50.3°,β16.7°,a26.7

17.

two triangles,  γ54.3°,β90.7°,b20.9  γ54.3°,β90.7°,b20.9or   γ 125.7°, β 19.3°, b 6.9   γ 125.7°, β 19.3°, b 6.9

19.

two triangles,  β75.7°, γ61.3°,b9.9  β75.7°, γ61.3°,b9.9or   β 18.3°, γ 118.7°, b 3.2   β 18.3°, γ 118.7°, b 3.2

21.

two triangles, α143.2°,β26.8°,a17.3 α143.2°,β26.8°,a17.3or α 16.8°, β 153.2°, a 8.3 α 16.8°, β 153.2°, a 8.3

23.

no triangle possible

25.

A47.8° A47.8°or A 132.2° A 132.2°

27.

8.6 8.6

29.

370.9 370.9

31.

12.3 12.3

33.

12.2  12.2 

35.

16.0  16.0 

37.

29.7° 29.7°

39.

x=76.9°or x=103.1° x=76.9°or x=103.1°

41.

110.6° 110.6°

43.

A39.4, C47.6, BC20.7  A39.4, C47.6, BC20.7 

45.

57.1 57.1

47.

42.0  42.0 

49.

430.2  430.2 

51.

10.1 10.1

53.

AD 13.8 AD 13.8

55.

AB2.8  AB2.8 

57.

L49.7, N56.3, LN5.8 L49.7, N56.3, LN5.8

59.

51.4 feet

61.

The distance from the satellite to station A Ais approximately 1716 miles. The satellite is approximately 1706 miles above the ground.

63.

2.6 ft

65.

5.6 km

67.

371 ft

69.

5936 ft

71.

24.1 ft

73.

19,056 ft2

75.

445,624 square miles

77.

8.65 ft2

10.2 Section Exercises

1.

two sides and the angle opposite the missing side.

3.

s sis the semi-perimeter, which is half the perimeter of the triangle.

5.

The Law of Cosines must be used for any oblique (non-right) triangle.

7.

11.3

9.

34.7

11.

26.7

13.

257.4

15.

not possible

17.

95.5°

19.

26.9°

21.

B45.9°,C99.1°,a6.4 B45.9°,C99.1°,a6.4

23.

A20.6°,B38.4°,c51.1 A20.6°,B38.4°,c51.1

25.

A37.8°,B43.8,C98.4° A37.8°,B43.8,C98.4°

27.

177.56 in2

29.

0.04 m2

31.

0.91 yd2

33.

3.0

35.

29.1

37.

0.5

39.

70.7°

41.

77.4°

43.

25.0

45.

9.3

47.

43.52

49.

1.41

51.

0.14

53.

18.3

55.

48.98

57.
A triangle. One angle is 52 degrees with opposite side = x. The other two sides are 5 and 6.
59.

7.62

61.

85.1

63.

24.0 km

65.

99.9 ft

67.

37.3 miles

69.

2371 miles

71.
Angle BO is 9.1 degrees, angle PH is 150.2 degrees, and angle DC is 20.7 degrees.
73.

599.8 miles

75.

65.4 cm2

77.

468 ft2

10.3 Section Exercises

1.

For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.

3.

Determine θ θfor the point, then move r runits from the pole to plot the point. If r ris negative, move r runits from the pole in the opposite direction but along the same angle. The point is a distance of r raway from the origin at an angle of θ θfrom the polar axis.

5.

The point ( 3, π 2 ) ( 3, π 2 )has a positive angle but a negative radius and is plotted by moving to an angle of π 2 π 2 and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point ( 3, π 2 ) ( 3, π 2 )has a negative angle and a positive radius and is plotted by first moving to an angle of π 2 π 2 and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.

7.

( 5,0 ) ( 5,0 )

9.

( 3 3 2 , 3 2 ) ( 3 3 2 , 3 2 )

11.

( 2 5 , 0.464 ) ( 2 5 , 0.464 )

13.

( 34 ,5.253 ) ( 34 ,5.253 )

15.

( 8 2 , π 4 ) ( 8 2 , π 4 )

17.

r=4cscθ r=4cscθ

19.

r= sinθ 2co s 4 θ 3 r= sinθ 2co s 4 θ 3

21.

r=3cosθ r=3cosθ

23.

r= 3sinθ cos( 2θ ) r= 3sinθ cos( 2θ )

25.

r= 9sinθ cos 2 θ r= 9sinθ cos 2 θ

27.

r= 1 9cosθsinθ r= 1 9cosθsinθ

29.

x 2 + y 2 =4x x 2 + y 2 =4xor ( x2 ) 2 4 + y 2 4 =1; ( x2 ) 2 4 + y 2 4 =1;circle

31.

3y+x=6; 3y+x=6;line

33.

y=3; y=3; line

35.

xy=4; xy=4;hyperbola

37.

x 2 + y 2 =4; x 2 + y 2 =4;circle

39.

x5y=3; x5y=3;line

41.

( 3, 3π 4 ) ( 3, 3π 4 )

43.

( 5,π ) ( 5,π )

45.
Polar coordinate system with a point located on the second concentric circle and two-thirds of the way between pi and 3pi/2 (closer to 3pi/2).
47.
Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi/2 and 2pi.
49.
Polar coordinate system with a point located on the fifth concentric circle and pi/2.
51.
Polar coordinate system with a point located on the third concentric circle and 2/3 of the way between pi/2 and pi (closer to pi).
53.
Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi/2.
55.

r= 6 5cosθsinθ r= 6 5cosθsinθ

Plot of given line in the polar coordinate grid
57.

r=2sinθ r=2sinθ

Plot of given circle in the polar coordinate grid
59.

r= 2 cosθ r= 2 cosθ

Plot of given circle in the polar coordinate grid
61.

r=3cosθ r=3cosθ

Plot of given circle in the polar coordinate grid.
63.

x 2 + y 2 =16 x 2 + y 2 =16

Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.
65.

y=x y=x

Plot of line y=x in the rectangular coordinates grid.
67.

x 2 + ( y+5 ) 2 =25 x 2 + ( y+5 ) 2 =25

Plot of circle with radius 5 centered at (0,-5).
69.

( 1.618,1.176 ) ( 1.618,1.176 )

71.

( 10.630,131.186° ) ( 10.630,131.186° )

73.

( 2,3.14 )or( 2,π ) ( 2,3.14 )or( 2,π )

75.

A vertical line with a aunits left of the y-axis. 

77.

A horizontal line with a aunits below the x-axis.

79.
Graph of shaded circle of radius 4 with the edge not included (dotted line) - polar coordinate grid.
81.
Graph of ray starting at (2, pi/4) and extending in a positive direction along pi/4 - polar coordinate grid.
83.
Graph of the shaded region 0 to pi/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid

10.4 Section Exercises

1.

Symmetry with respect to the polar axis is similar to symmetry about the x x -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line θ= π 2 θ= π 2 is similar to symmetry about the y y -axis.

3.

Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at θ=0, π 2 ,πand  3π 2 , θ=0, π 2 ,πand  3π 2 , and sketch the graph.

5.

The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

7.

symmetric with respect to the polar axis

9.

symmetric with respect to the polar axis, symmetric with respect to the line θ= π 2 , θ= π 2 , symmetric with respect to the pole

11.

no symmetry

13.

no symmetry

15.

symmetric with respect to the pole

17.

circle

Graph of given circle.
19.

cardioid

Graph of given cardioid.
21.

cardioid

Graph of given cardioid.
23.

one-loop/dimpled limaçon

Graph of given one-loop/dimpled limaçon
25.

one-loop/dimpled limaçon

Graph of given one-loop/dimpled limaçon
27.

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
29.

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
31.

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
33.

lemniscate

Graph of given lemniscate (along horizontal axis)
35.

lemniscate

Graph of given lemniscate (along y=x)
37.

rose curve

Graph of given rose curve - four petals.
39.

rose curve

Graph of given rose curve - eight petals.
41.

Archimedes’ spiral

Graph of given Archimedes' spiral
43.

Archimedes’ spiral

Graph of given Archimedes' spiral
45.
Graph of given equation.
47.
Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)
49.
Graph of given equation.
51.
Graph of given equation. Similar to original Archimedes' spiral.
53.
Graph of given equation.
55.

They are both spirals, but not quite the same.

57.

Both graphs are curves with 2 loops. The equation with a coefficient of θ θ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to 4π 4π to get a better picture.

59.

When the width of the domain is increased, more petals of the flower are visible.

61.

The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

63.

The graphs are spirals. The smaller the coefficient, the tighter the spiral.

65.

( 4, π 3 ),( 4, 5π 3 ) ( 4, π 3 ),( 4, 5π 3 )

67.

( 3 2 , π 3 ),( 3 2 , 5π 3 ) ( 3 2 , π 3 ),( 3 2 , 5π 3 )

69.

( 0, π 2 ),( 0,π ),( 0, 3π 2 ),( 0,2π ) ( 0, π 2 ),( 0,π ),( 0, 3π 2 ),( 0,2π )

71.

( 8 4 2 , π 4 ),( 8 4 2 , 5π 4 ) ( 8 4 2 , π 4 ),( 8 4 2 , 5π 4 ) and at θ= 3π 4 , 7π 4 θ= 3π 4 , 7π 4 since r r is squared

10.5 Section Exercises

1.

a is the real part, b is the imaginary part, and i= 1 i= 1

3.

Polar form converts the real and imaginary part of the complex number in polar form using x=rcosθ x=rcosθ and y=rsinθ. y=rsinθ.

5.

z n = r n ( cos( nθ )+isin( nθ ) ) z n = r n ( cos( nθ )+isin( nθ ) )It is used to simplify polar form when a number has been raised to a power.

7.

5 2 5 2

9.

38 38

11.

14.45 14.45

13.

4 5 cis( 333.4° ) 4 5 cis( 333.4° )

15.

2cis( π 6 ) 2cis( π 6 )

17.

7 3 2 +i 7 2 7 3 2 +i 7 2

19.

2 3 2i 2 3 2i

21.

1.5i 3 3 2 1.5i 3 3 2

23.

4 3 cis( 198° ) 4 3 cis( 198° )

25.

3 4 cis( 180° ) 3 4 cis( 180° )

27.

5 3 cis( 17π 24 ) 5 3 cis( 17π 24 )

29.

7cis( 70° ) 7cis( 70° )

31.

5cis( 80° ) 5cis( 80° )

33.

5cis( π 3 ) 5cis( π 3 )

35.

125cis( 135° ) 125cis( 135° )

37.

9cis( 240° ) 9cis( 240° )

39.

cis( 3π 4 ) cis( 3π 4 )

41.

3cis( 80° ),3cis( 200° ),3cis( 320° ) 3cis( 80° ),3cis( 200° ),3cis( 320° )

43.

2 4 3 cis( 2π 9 ),2 4 3 cis( 8π 9 ),2 4 3 cis( 14π 9 ) 2 4 3 cis( 2π 9 ),2 4 3 cis( 8π 9 ),2 4 3 cis( 14π 9 )

45.

2 2 cis( 7π 8 ),2 2 cis( 15π 8 ) 2 2 cis( 7π 8 ),2 2 cis( 15π 8 )

47.
Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).
49.
Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).
51.
Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).
53.
Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).
55.
Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).
57.

3.61 e 0.59i 3.61 e 0.59i

59.

2+3.46i 2+3.46i

61.

4.332.50i 4.332.50i

10.6 Section Exercises

1.

A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, x=f( t ) x=f( t )and y=f( t ). y=f( t ).

3.

Choose one equation to solve for t, t,substitute into the other equation and simplify.

5.

Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.

7.

y=2+2x y=2+2x

9.

y=3 x1 2 y=3 x1 2

11.

x=2 e 1y 5 x=2 e 1y 5 or y=15ln( x 2 ) y=15ln( x 2 )

13.

x=4log( y3 2 ) x=4log( y3 2 )

15.

x= ( y 2 ) 3 y 2 x= ( y 2 ) 3 y 2

17.

y= x 3 y= x 3

19.

( x 4 ) 2 + ( y 5 ) 2 =1 ( x 4 ) 2 + ( y 5 ) 2 =1

21.

y 2 =1 1 2 x y 2 =1 1 2 x

23.

y= x 2 +2x+1 y= x 2 +2x+1

25.

y= ( x+1 2 ) 3 2 y= ( x+1 2 ) 3 2

27.

y=3x+14 y=3x+14

29.

y=x+3 y=x+3

31.

{ x( t )=t y( t )=2sint+1 { x( t )=t y( t )=2sint+1

33.

{ x( t )= t +2t y( t )=t { x( t )= t +2t y( t )=t

35.

{ x( t )=4cost y( t )=6sint ; { x( t )=4cost y( t )=6sint ;Ellipse

37.

{ x( t )= 10 cost y( t )= 10 sint ; { x( t )= 10 cost y( t )= 10 sint ; Circle

39.

{ x( t )=1+4t y( t )=2t { x( t )=1+4t y( t )=2t

41.

{ x( t )=4+2t y( t )=13t { x( t )=4+2t y( t )=13t

43.

yes, at t=2 t=2

45.

t t x x y y
1 -3 1
2 0 7
3 5 17
47.

answers may vary: { x( t )=t1 y( t )= t 2  and { x( t )=t+1 y( t )= ( t+2 ) 2 { x( t )=t1 y( t )= t 2  and { x( t )=t+1 y( t )= ( t+2 ) 2

49.

answers may vary: , { x( t )=t y( t )= t 2 4t+4  and { x( t )=t+2 y( t )= t 2 { x( t )=t y( t )= t 2 4t+4  and { x( t )=t+2 y( t )= t 2

10.7 Section Exercises

1.

plotting points with the orientation arrow and a graphing calculator

3.

The arrows show the orientation, the direction of motion according to increasing values of t. t.

5.

The parametric equations show the different vertical and horizontal motions over time.

7.
Graph of the given equations - looks like an upward opening parabola.
9.
Graph of the given equations - a line, negative slope.
11.
Graph of the given equations - looks like a sideways parabola, opening to the right.
13.
Graph of the given equations - looks like the left half of an upward opening parabola.
15.
Graph of the given equations - looks like a downward opening absolute value function.
17.
Graph of the given equations - a vertical ellipse.
19.
Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.
21.
Graph of the given equations- looks like an upward opening parabola.
23.
Graph of the given equations- looks like a downward opening parabola.
25.
Graph of the given equations- horizontal ellipse.
27.
Graph of the given equations- looks like the lower half of a sideways parabola opening to the right
29.
Graph of the given equations- looks like an upwards opening parabola
31.
Graph of the given equations- looks like the upper half of a sideways parabola opening to the left
33.
Graph of the given equations- the left half of a hyperbola with diagonal asymptotes
35.
Graph of the given equations - vertical periodic trajectory
37.
Graph of the given equations - vertical periodic trajectory
39.

There will be 100 back-and-forth motions.

41.

Take the opposite of the x( t ) x( t ) equation.

43.

The parabola opens up.

45.

{ x( t )=5cost y( t )=5sint { x( t )=5cost y( t )=5sint

47.
Graph of the given equations
49.
Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 1 unit.
51.
Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 3 units.
53.

a=4,b=3,c=6,d=1 a=4,b=3,c=6,d=1

55.

a=4,b=2,c=3,d=3 a=4,b=2,c=3,d=3

57.
Graph of the given equations Graph of the given equations Graph of the given equations
59.
Graph of the given equations Graph of the given equations Graph of the given equations
61.

The y y -intercept changes.

63.

y( x )=16 ( x 15 ) 2 +20( x 15 ) y( x )=16 ( x 15 ) 2 +20( x 15 )

65.

{ x(t)=64tcos( 52° ) y(t)=16 t 2 +64tsin( 52° ) { x(t)=64tcos( 52° ) y(t)=16 t 2 +64tsin( 52° )

67.

approximately 3.2 seconds

69.

1.6 seconds

71.
Graph of the given equations - a hypocycloid
73.
Graph of the given equations - a four petal rose

10.8 Section Exercises

1.

lowercase, bold letter, usually u,v,w u,v,w

3.

They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

5.

The first number always represents the coefficient of the i, i,and the second represents the j. j.

7.

7,5 7,5

9.

not equal

11.

equal

13.

equal

15.

7i3j 7i3j

17.

6i2j 6i2j

19.

u+v= 5,5 ,uv= 1,3 ,2u3v= 0,5 u+v= 5,5 ,uv= 1,3 ,2u3v= 0,5

21.

10i4j 10i4j

23.

2 29 29 i+ 5 29 29 j 2 29 29 i+ 5 29 29 j

25.

2 229 229 i+ 15 229 229 j 2 229 229 i+ 15 229 229 j

27.

7 2 10 i+ 2 10 j 7 2 10 i+ 2 10 j

29.

| v |=7.810,θ=39.806° | v |=7.810,θ=39.806°

31.

| v |=7.211,θ=236.310° | v |=7.211,θ=236.310°

33.

6 6

35.

12 12

37.
39.
Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).
41.
Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).
43.
Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).
45.
Vector extending from the origin to (7,5), taking the base as the origin.
47.

4,1 4,1

49.

v=7i+3j v=7i+3j

Vector going from (4,-1) to (-3,2).
51.

3 2 i+3 2 j 3 2 i+3 2 j

53.

i 3 j i 3 j

55.

a. 58.7; b. 12.5

57.

x=7.13 x=7.13pounds, y=3.63 y=3.63pounds

59.

x=2.87 x=2.87pounds, y=4.10 y=4.10pounds

61.

4.635 miles, 17.764° N of E

63.

17 miles. 10.318 miles

65.

Distance: 2.868. Direction: 86.474° North of West, or 3.526° West of North

67.

4.924°. 659 km/hr

69.

4.424°

71.

( 0.081,8.602 ) ( 0.081,8.602 )

73.

21.801°, relative to the car’s forward direction

75.

parallel: 16.28, perpendicular: 47.28 pounds

77.

19.35 pounds, 231.54° from the horizontal

79.

5.1583 pounds, 75.8° from the horizontal

Review Exercises

1.

Not possible

3.

C=120°,a=23.1,c=34.1 C=120°,a=23.1,c=34.1

5.

distance of the plane from point A: A:2.2 km, elevation of the plane: 1.6 km

7.

B=71.0°,C=55.0°,a=12.8 B=71.0°,C=55.0°,a=12.8

9.

40.6 km

11.


Polar coordinate grid with a point plotted on the fifth concentric circle 2/3 the way between pi and 3pi/2 (closer to 3pi/2).
13.

( 0,2 ) ( 0,2 )

15.

( 9.8489,203.96° ) ( 9.8489,203.96° )

17.

r=8 r=8

19.

x 2 + y 2 =7x x 2 + y 2 =7x

21.

y=x y=x

Plot of the function y=-x in rectangular coordinates.
23.

symmetric with respect to the line θ= π 2 θ= π 2

25.


Graph of the given polar equation - an inner loop limaçon.
27.


Graph of the given polar equation - a cardioid.
29.

5

31.

cis( π 3 ) cis( π 3 )

33.

2.3+1.9i 2.3+1.9i

35.

60cis( π 2 ) 60cis( π 2 )

37.

3cis( 4π 3 ) 3cis( 4π 3 )

39.

25cis( 3π 2 ) 25cis( 3π 2 )

41.

5cis( 3π 4 ),5cis( 7π 4 ) 5cis( 3π 4 ),5cis( 7π 4 )

43.
Plot of -1 + 3i in the complex plane (-1 along the real axis, 3 along the imaginary).
45.

x 2 + 1 2 y=1 x 2 + 1 2 y=1

47.

{ x( t )=2+6t y( t )=3+4t { x( t )=2+6t y( t )=3+4t

49.

y=2 x 5 y=2 x 5

Plot of the given parametric equations.
51.
  1. { x( t )=( 80cos( 40° ) )t y( t )=16 t 2 +( 80sin( 40° ) )t+4 { x( t )=( 80cos( 40° ) )t y( t )=16 t 2 +( 80sin( 40° ) )t+4
  2. The ball is 14 feet high and 184 feet from where it was launched.
  3. 3.3 seconds
53.

not equal

55.

4i

57.

3 10 10 3 10 10 i 10 10 10 10 j

59.

Magnitude: 3 2 , 3 2 ,Direction: 225° 225°

61.

16 16

63.


Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).

Practice Test

1.

α=67.1°,γ=44.9°,a=20.9 α=67.1°,γ=44.9°,a=20.9

3.

1712 miles 1712 miles

5.

( 1, 3 ) ( 1, 3 )

7.

y=3 y=3

Plot of the given equation in rectangular form - line y=-3.
9.


Graph of the given equations - a cardioid.
11.

106 106

13.

5 2 +i 5 3 2 5 2 +i 5 3 2

15.

4cis( 21° ) 4cis( 21° )

17.

2 2 cis( 18° ),2 2 cis( 198° ) 2 2 cis( 18° ),2 2 cis( 198° )

19.

y=2 ( x1 ) 2 y=2 ( x1 ) 2

21.


Graph of the given equations - a vertical ellipse.
23.

−4i − 15j

25.

2 13 13 i+ 3 13 13 j 2 13 13 i+ 3 13 13 j

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