Trigonometry/Calculus – Curriculum Map

Marking Connection to text CCSS Vocabulary Period Trigonometry with What are the State Standards students will be What vocabulary will support the Applications (1983) learning? students’ learning?

Chapter 1  Angle Trigonometric Functions Geometry  Vertex BASIC CONCEPTS  Degrees Sec.1-1: Angles and Degree Similarity, right triangles, and trigonometry  Minutes Measure Define trigonometric ratios and solve  Seconds problems involving right triangles  Initial Side THE TRIGONOMETRIC G.SRT.6 Understand that by similarity, side  Terminal Side FUNCTIONS ratios in right triangles are properties of the  Directed Angle Sec.1-2: Sine and Cosine angles in the triangle, leading to definitions of  Standard Position

trigonometric ratios for acute angles.  Coterminal Angles Sec.1-3: Values of the Sine and Cosine Functions  Sine Function G.SRT.7 Explain and use the relationship  Cosine Function between the sine and cosine of complementary Sec.1-4: Other Trigonometric  Significant Digit angles. st Functions  Scientific Notation 1  Tangent Function Marking Sec.1-5: Solving Right G.SRT.8 Use trigonometric ratios and the  Secant Period Triangles Pythagorean Theorem to solve right triangles in  Cosecant applied problems.  Cotangent Sec.1-6: Trigonometric  Angle of Elevation Functions of Arbitrary Angles  Angle of Depression  Quadrantal Angles

 Reference Triangle  Reference Angle

Chapter 2 Circles Circular Functions and Their  Radian Measure Graphs Find arc lengths and areas of sectors of  Semi-Circle circles

CIRCULAR FUNCTIONS G.C.5 Derive using similarity the fact that the

Sec.2-1: Radian Measure length of the arc intercepted by an angle is

proportional to the radius, and define the radian

measure of the angle as the constant of

proportionality; derive the formula for the area

of a sector

Trigonometric Functions Sec.2-2: Circular Functions Extend the domain of trigonometric functions using the unit circle Sec.2-3: Uniform Circular F.TF.1 Understand radian measure of an angle  Circular Functions Motion as the length of the arc on the unit circle  Unit Circle

subtended by the angle.  Uniform Circular Motion  Linear Speed

GRAPHS  Angular Speed F.TF.2 Explain how the unit circle in the Sec.2-4: Graphing the Circular  Even Functions Functions coordinate plane enables the extension of  Odd Functions

trigonometric functions to all real numbers,  Asymptote

interpreted as radian measures of angles  Periodicity INVERSE CIRCULAR traversed counterclockwise around the unit  Fundamental Period FUNCTIONS circle. Sec.2-5: The Inverse Circular  Inverse Function F.TF.3 Use special triangles to determine Functions  One-to-One geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number

F.TF.4 Use the unit circle to explain symmetry

(odd and even) and periodicity of trigonometric functions. st 1

Marking Model periodic phenomena with Period trigonometric functions F.TF.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows

its inverse to be constructed

F.TF.7 Use inverse functions to solve trigonometric equations that arise in modeling

contexts; evaluate the solutions using

technology, and interpret them in terms of the context.

Chapter 3 Properties of Trigonometric

Functions Trigonometric Functions Prove and apply trigonometric identities TRIGONOMETRIC IDENTITIES Sec.3-1: Simplifying  Identities

Trigonometric Expressions F.TF.8 Prove the Pythagorean identity sin2(θ) +  Pythagorean Identities cos2(θ) = 1 and use it to find sin(θ), cos(θ), or  Sum and Difference Sec.3-2: Proving Identities tan(θ) given sin(θ), cos(θ), or tan(θ) and the Identities quadrant of the angle  Double Angle Identities Sec.3-3: Trigonometric Sum  Half-Angle Identities and Difference Formulas  Trigonometric Equation

Sec.3-4: Double-Angle and

Half-Angle Formulas

TRIGONOMETRIC EQUATIONS F.TF.9 Prove the addition and subtraction Sec.3-5: Solving Trigonometric formulas for sine, cosine, and tangent and use Equations them to solve problems. Sec.3-6: The Specialized

Identity

Chapter 4 Geometry Oblique Triangles Apply trigonometry to general triangles

 Law of Cosines G.SRT.10 Prove the Laws of Sines and Cosines SOLVING OBLIQUE  Oblique Triangles TRIANGLES and use them to solve problems.  Law of Sines Sec.4-1: The Law of Cosines st  Ambiguous Case 1 G.SRT.11 Understand and apply the Law of  Area of an Oblique Marking Sec.4-2: The Law of Sines Sines and the Law of Cosines to find unknown Triangle Period measurements in right and non-right triangles  SAS Sec.4-3: The Ambiguous Case (e.g., surveying problems, resultant forces).  ASA

 SSS

 AAS

AREA FORMULAS FOR G.SRT.9 Derive the formula A = 1/2 ab sin(C)  Hero’s Formula OBLIQUE TRIANGLES for the area of a triangle by drawing an auxiliary Sec.4-4: The Area of an line from a vertex perpendicular to the opposite Oblique Triangle side

Marking Connection to text CCSS Vocabulary Period Trigonometry with What are the State Standards students will be What vocabulary will support the Applications (1983) learning? students’ learning?

Chapter 5 Sinusoidal Variations Trigonometric Functions  Oscillate GRAPHING SINUSOIDS Extend the domain of trigonometric functions  Sinusoid Sec.5-1: Period and Amplitude using the unit circle  Varies Sinusoidally  Amplitude Sec.5-2: Phase Shift and F.TF.2 Explain how the unit circle in the  Period Vertical Shift coordinate plane enables the extension of  Fundamental Period

trigonometric functions to all real numbers,  Translation Sec.5-3: Simple Harmonic  Phase Shift Motion interpreted as radian measures of angles traversed  Vertical Shift counterclockwise around the unit circle GRAPHING COMBINATIONS  Simple Harmonic Motion OF SINUSOIDS F.TF.5 Choose trigonometric functions to model  Frequency Sec.5-4: Graphing by Addition of periodic phenomena with specified amplitude,  Fundamental Ordinates frequency, and midline.  Overtones

2nd Chapter 6 Marking Vectors in the Plane Vector and Matrix Quantities Period VECTOR OPERATIONS AND Represent and model with vector quantities  Scalar quantities APPLICATIONS  Scalar N.VM.1 Recognize vector quantities as having  Vector quantities Sec.6-1: Basic Vector Operations both magnitude and direction. Represent vector  Initial point quantities by directed line segments, and use  Terminal point Sec.6-2: Vectors and Navigation appropriate symbols for vectors and their  Vector Addition

magnitudes (e.g., v, | v|, || v||, v).  Vector Multiplication Sec.6-3: Vectors and Force  Linear Combination N.VM.2 Find the components of a vector by  Standard Position subtracting the coordinates of an initial point  Scalar Components from the coordinates of a terminal point.  Norm

N.VM.3 Solve problems involving velocity and  Unit Vector

other quantities that can be represented by vectors  Displacement

N.VM.4.A Add vectors end-to-end, component-  Bearing wise, and by the parallelogram rule. Understand  Heading that the magnitude of a sum of two vectors is  Ground Speed typically not the sum of the magnitudes.  Air Speed N.VM.4.B Given two vectors in magnitude and  True Course direction form, determine the magnitude and  Newton direction of their sum  Tension N.VM.4.C Understand vector subtraction v – w as v + (– w), where – w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector

subtraction graphically by connecting the tips in

the appropriate order, and perform vector subtraction component-wise.

THE DOT PRODUCT AND ITS N.VM.5.A Represent scalar multiplication  Dot Product APPLICATIONS graphically by scaling vectors and possibly  Orthogonal Sec.6-4: The Dot Product reversing their direction; perform scalar  Parallel multiplication component-wise, e.g., as c (v x, v  Scalar Projection Sec.6-5: Vectors, Work, and y) = (cv x, cv y).  Work Energy N.VM.5.B Compute the magnitude of a scalar  Energy multiple c v using || c v|| = | c | v. Compute the  Joule (J)

 Kilowatt-hour (kW*h) direction of c v knowing that when | c |v ≠ 0, the

direction of c v is either along v (for c > 0) or

against v (for c < 0)

Chapter 7 The Complex Number System Complex Numbers Perform arithmetic operations with complex  Pole POLAR COORDINATES nd numbers. 2 Marking Sec.7-1: The Polar Coordinate  Polar Axis Period System N.CN.1 Know there is a complex number i such  Polar Coordinates that i 2 = –1, and every complex number has the  Rectangular Sec.7-2: Graphs of Polar form a + bi with a and b real. Coordinates Equations N.CN.2 Use the relation i 2 = –1 and the  Distance Formula commutative, associative, and distributive  Three-Leaved Rose Sec.7-3: Conic Sections properties to add, subtract, and multiply complex  Lemniscate numbers.  Limacon REPRESENTING COMPLEX N.CN.3 Find the conjugate of a complex number;  Cardioid NUMBERS use conjugates to find moduli and quotients of  Conic Section Sec.7-4: Complex Numbers  Eccentricity complex numbers.  Focus

Sec.7-5: The Complex Plane and  Directrix the Polar Form Represent complex numbers and their  Ellipse operations on a complex plane.  Complex Number USING THE POLAR FORM OF N.CN.4 Represent complex numbers on the  Imaginary Part COMPLEX NUMBERS complex plane in rectangular and polar form  Real Part Sec.7-6: The Polar Form of (including real and imaginary numbers), and  Conjugate Products and Quotients explain why the rectangular and polar forms of a  Modulus (Absolute

given complex number represent the same Value) Sec.7-7: De Moivre’s Theorem number.  Complex(Argand) Plane Sec.7-8: Roots of Complex N.CN.5 Represent addition, subtraction,  Real Axis Numbers multiplication, and conjugation of complex  Imaginary Axis numbers geometrically on the complex plane; use  Polar Form properties of this representation for computation.  Argument For example, (–1 + √3 i)3 = 8 because (–1 + √3 i)  De Moivre’s Theorem has modulus 2 and argument 120°.  nth Root N.CN.6 Calculate the distance between numbers

in the complex plane as the modulus of the

difference, and the midpoint of a segment as the average of the numbers at its endpoints

Chapter 8 Congruence Transformations Experiment with transformations in the plane  Translation Sec.8-1: Translations G.CO.2 Represent transformations in the plane  Rotation

using, e.g., transparencies and geometry software;  Alias Sec.8-2: Rotations  Alibi describe transformations as functions that take  Standard Form of a Sec.8-3: Alias versus Alibi points in the plane as inputs and give other points Second-Degree as outputs. Compare transformations that Equation Sec.8-4: Second-Degree preserve distance and angle to those that do not Equations (e.g., translation versus horizontal stretch). G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or

geometry software. Specify a sequence of transformations that will carry a given figure onto

another

nd 2 Marking Period

Marking Connection to text CCSS Vocabulary Period Trigonometry with What are the State Standards students will be What vocabulary will support the Applications (1983) learning? students’ learning?

Chapter 9 Vectors in Space Vector and Matrix Quantities  xz-plane COORDINATES AND Represent and model with vector quantities  Coordinate Planes VECTORS  Octants Sec.9-1: Rectangular N.VM.1 Recognize vector quantities as having  Sphere Coordinates in Space both magnitude and direction. Represent vector  Origin quantities by directed line segments, and use  Position Vector Sec.9-2: Vectors in Space appropriate symbols for vectors and their  Vector Equation of a Line LINES AND PLANES magnitudes (e.g., v, | v|, || v||, v).  Normal Vector Sec.9-3: Lines in Space N.VM.2 Find the components of a vector by  Traces subtracting the coordinates of an initial point  Intercepts Sec.9-4: Planes from the coordinates of a terminal point.

N.VM.3 Solve problems involving velocity and

other quantities that can be represented by vectors

N.VM.4.A Add vectors end-to-end, component-

rd wise, and by the parallelogram rule. Understand 3 that the magnitude of a sum of two vectors is Marking Period typically not the sum of the magnitudes.

N.VM.4.B Given two vectors in magnitude and

direction form, determine the magnitude and

direction of their sum N.VM.4.C Understand vector subtraction v – w

as v + (– w), where – w is the additive inverse of

w, with the same magnitude as w and pointing in the opposite direction. Represent vector

subtraction graphically by connecting the tips in

the appropriate order, and perform vector

subtraction component-wise.

.

Chapter 10  Great circle Spherical Trigonometry Geometric Measurement and Dimension  Meridians SPHERICAL GEOMETRY Visualize relationships between two-  Parallels Sec.10-1: Some Spherical dimensional and three-dimensional objects.  Spherical Angle Geometry  Spherical Triangle G.GMD.4 Identify the shapes of two-  Poles SOLVING RIGHT SPHERICAL  Polar Triangle TRIANGLES dimensional cross-sections of threedimensional  Pythagorean Theorem Sec.10-2: Right Spherical objects, and identify three-dimensional objects  Napier’s Rule Triangles generated by rotations of two-dimensional  Circular Parts objects. Sec.10-3: Napier’s Rules  Middle Parts  Adjacent Parts Apply geometric concepts in modeling  Opposite Parts situations.  Quadrantal Triangle SOLVING OBLIQUE SPHERICAL TRIANGLES  Oblique Spherical G.MG.1 Use geometric shapes, their measures, Sec.10-4: Oblique Spherical Triangle Triangles and their properties to describe objects.  Law of Sines  Law of Cosines for Sec.10-5: Great-Circle Sides 3rd Marking Navigation  Law of Cosines for Period Angles  Latitude, Longitude  Terrestrial Triangle  Nautical Mile

Chapter 11 Building Functions  Power Series Infinite Series Build a function that models a relationship  Taylor Series Sec.11-1: Power Series between two quantities.  Hyperbolic Sine

F.BF.1.A Determine an explicit expression, a  Hyperbolic Cosine Sec.11-2: Hyperbolic Functions recursive process, or steps for calculation from a  Fourier Series

context. Sec.11-3: Trigonometric Series F.BF.2 Write arithmetic and geometric sequences

both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Marking Connection to text CCSS Vocabulary Period Multiple Online Sources What are the State Standards students will be What vocabulary will support the (No Text) learning? students’ learning?

 Functions Introduction to Calculus  Continuous Functions Functions and Limits While there may be correlations to the CCSS’s,  Discontinuous the preview of Calculus is intended to follow Functions Sec.1: Continuous and the College Board Curriculum Framework for  Limits discontinuous Functions Calculus.  Limit of a Sum  Limit of a Product Sec.2: Limits of continuous  Limit of a Quotient functions  Increments

Sec.3: Limits of discontinuous

functions

Definition of the Derivative  Delta (Change in)

 Secants Sec.1: Tangent Lines and General Rates of Change  Tangents  Slope Sec.2: Using the Delta Method to Differentiate

4th Sec.3: Finding the Slope of a Marking Curve at a given point Period Differentiation of Functions While there may be correlations to the CCSS’s, SPECIAL RULES the preview of Calculus is intended to follow  The Power Rule the College Board Curriculum Framework for  Constant Multiple Rule Sec.1: Techniques of Calculus.  Sum and Difference Differentiation Rules  Chain Rule Sec.2: The Chain Rule  Product Rule  Quotient Rule Sec.3: The Product Rule

Sec.4: The Quotient Rule

Application of the Derivative  Increasing  Decreasing Sec.1: Determining if a curve is  Concavity rising, falling, or critical  Relative Maximum  Relative Minimum Sec.2: Determining if a critical  Critical Points point is a relative max or min  Maximum Area

 Maximum Volume Sec.3: Apply the derivative to  Rate of Change area and volume problems  Average Velocity Sec.4: Apply the derivative to  Instantaneous Velocity Distance-Rate-Time problems

Marking Connection to text CCSS Vocabulary Period Multiple Online Sources What are the State Standards students will be What vocabulary will support the (No Text) learning? students’ learning?

Anti-Derivative Integrals While there may be correlations to the CCSS’s,  Anti-Derivative the preview of Calculus is intended to follow  Antidifferentiation Sec.1: Anti-Derivative the College Board Curriculum Framework for  Indefinite Integral Calculus.  Constant of Integration Sec.2: Integration of Sums  Integration of a Sum  Integration by Sec.3: Integration by Substitution Substitution

Definite Integrals AREA UNDER A CURVE  Riemann Sum Sec.1: Riemann Sum  Rectangle Method

 Trapezoid Method Sec2: Finding Area Under a Curve Using Definite Integral  Definite Integral

4th Marking Period