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Trigonometry/Calculus – Curriculum Map

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Trigonometry/Calculus – Curriculum Map 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 Vertex BASIC CONCEPTS Geometry 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 Circular Functions and Their Circles 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 Unit Circle as the length of the arc on the unit circle Uniform Circular Motion subtended by the angle. 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 FUNCTIONS Fundamental Period 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 TRIGONOMETRIC IDENTITIES Prove and apply 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 Half-Angle Identities Sec.3-3: Trigonometric Sum 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.
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