Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 6 Applications of Trigonometry
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.1 Vectors in the Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 What you’ll learn about Two-Dimensional Vectors Vector Operations Unit Vectors Direction Angles Applications of Vectors … and why These topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 Directed Line Segment
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Two-Dimensional Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Initial Point, Terminal Point, Equivalent
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Magnitude
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Magnitude of a Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Magnitude of a Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Vector Addition and Scalar Multiplication
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Performing Vector Operations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Performing Vector Operations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Unit Vectors
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding a Unit Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding a Unit Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Standard Unit Vectors
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Resolving the Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Components of a Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Components of a Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Direction Angle of a Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Direction Angle of a Vector
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Velocity and Speed The velocity of a moving object is a vector because velocity has both magnitude and direction. The magnitude of velocity is speed.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.2 Dot Product of Vectors
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about The Dot Product Angle Between Vectors Projecting One Vector onto Another Work … and why Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Dot Product
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. 1. u·v=v·u 2. u·u=|u| ·u=0 4. u·(v+w)=u·v+u·w (u+v) ·w=u·w+v·w 5. (cu) ·v=u·(cv)=c(u·v)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Dot Product
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Dot Product
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Angle Between Two Vectors
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Angle Between Vectors
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Angle Between Vectors
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Projection of u and v
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Work
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.3 Parametric Equations and Motion
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher … and why These topics can be used to model the path of an object such as a baseball or golf ball.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Parametric Curve, Parametric Equations The graph of the ordered pairs (x,y) where x = f(t) and y = g(t) are functions defined on an interval I of t-values is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Graphing Parametric Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Graphing Parametric Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Eliminating the Parameter
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Eliminating the Parameter
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Eliminating the Parameter
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Eliminating the Parameter
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Parametric Equations for a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Parametric Equations for a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.4 Polar Coordinates
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Use the Law of Cosines to find the measure of the third side of the given triangle º º 6 11
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions Use the Law of Cosines to find the measure of the third side of the given triangle º º
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Polar Coordinate System Coordinate Conversion Equation Conversion Finding Distance Using Polar Coordinates … and why Use of polar coordinates sometimes simplifies complicated rectangular equations and they are useful in calculus.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Polar Coordinate System
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Plotting Points in the Polar Coordinate System
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Plotting Points in the Polar Coordinate System
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Finding all Polar Coordinates of a Point
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Coordinate Conversion Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Polar to Rectangular Coordinates
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Polar to Rectangular Coordinates
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Rectangular to Polar Coordinates
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Rectangular to Polar Coordinates
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Polar Form to Rectangular Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Polar Form to Rectangular Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Polar Form to Rectangular Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Converting from Polar Form to Rectangular Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.5 Graphs of Polar Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Polar Curves and Parametric Curves Symmetry Analyzing Polar Curves Rose Curves Limaçon Curves Other Polar Curves … and why Graphs that have circular or cylindrical symmetry often have simple polar equations, which is very useful in calculus.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Symmetry The three types of symmetry figures to be considered will have are: 1. The x-axis (polar axis) as a line of symmetry. 2. The y-axis (the line θ = π/2) as a line of symmetry. 3. The origin (the pole) as a point of symmetry.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Symmetry Tests for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation. To Test for SymmetryReplaceBy 1. about the x-axis(r,θ) (r,-θ) or (-r, π-θ) 2. about the y-axis(r,θ) (-r,-θ) or (r, π-θ) 3. about the origin(r,θ) (-r,θ) or (r, π+θ)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Testing for Symmetry
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Testing for Symmetry
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Rose Curves
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Limaçon Curves
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.6 De Moivre’s Theorem and nth Roots
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about The Complex Plane Trigonometric Form of Complex Numbers Multiplication and Division of Complex Numbers Powers of Complex Numbers Roots of Complex Numbers … and why The material extends your equation-solving technique to include equations of the form z n = c, n is an integer and c is a complex number.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Complex Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Absolute Value (Modulus) of a Complex Number
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Graph of z = a + bi
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Trigonometric Form of a Complex Number
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Trigonometric Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Trigonometric Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Product and Quotient of Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Multiplying Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Multiplying Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A Geometric Interpretation of z 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide De Moivre’s Theorem
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using De Moivre’s Theorem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide nth Root of a Complex Number
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Finding nth Roots of a Complex Number
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Cube Roots
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Cube Roots
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions