# Trigonometry/Calculus

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1 /Calculus Dunmore School District Dunmore, PA

2 /Calculus Prerequisite: Successful completion of Algebra II In the first half of the course students will study., helps students develop skills sufficiently to write and use the definition of trigonometric functions; solve right and oblique triangles; use the unit circle; learn and apply the laws of sine and cosine; prove identities; solve trigonometric equations; and sketch the graphs of the trigonometric functions. The second half of the course is designed to introduce students to calculus. Students will begin by studying relations and functions. They will then study linear, polynomial, absolute value, square root, logarithmic, exponential, reciprocal and greatest integer functions. For each function they will learn domain, range, graphing and graph properties, followed by applications that apply to that function. The year will end with an introduction to derivatives and integrals where students will learn limits, power rules and applications. /Calculus Page 1

3 Year-at-a-glance Subject: /Calculus Grade Level: 12 Date Completed: 1/26/17 1 st Quarter with Right Triangles Unit Circle Definition of Trigonometric Topic Resources Standards Chapter 5 Sections 2-4 p Chapter 6 Sections 1-2 p CC.2.3.HS.A.7 G G HSG.SRT.C.6 HSG.SRT.C.7 HSG.SRT.C.8 HSF.TF.A.1 HSF.TF.A.2 HSF.TF.A.3 CC.2.2.HS.C.7 Graphs of Trigonometric Chapter 6 Sections 3-6 p HSF.TF.A.4 HSF.TF.B.5 CC.2.2.HS.C.8 /Calculus Page 2

4 Trigonometric Algebra Dunmore School District Chapter 7 Sections 1-4, 6-7 p , CC.2.2.HS.C.9 G G G HSF.TF.C.8 HSF.TF.C.9 /Calculus Page 3

5 2 nd Quarter Continue Trigonometric Algebra Topic Resources Standards Chapter 7 Sections 1-4, 6-7 p , CC.2.2.HS.C.9 G G G HSF.TF.C.8 HSF.TF.C.9 with General Triangles Introduction to Chapter 8 Sections 1-2 p Chapter 2 Section 3 p HSG.SRT.D.10 HSG.SRT.D.11 HSG.SRT.D.9 CC C.1 /Calculus Page 4

6 3 rd Quarter Linear Polynomial Absolute Value Topic Resources Standards Chapter 2 Section 4 p Chapter 3 Sections 1, 3-4 p , Chapter 1 Section 8 p CC.2.2.HS.C.5 CC.2.4.HS.B.3 CC.2.2.HS.C.5 Square Root Review Chapter Section 7 p Chapter 1 Section 6 p Chapter 2 Section 6 p.250 /Calculus Page 5

7 Logarithmic Dunmore School District Chapter 4 Sections 3-5 p /Calculus Page 6

8 4 th Quarter Exponential Reciprocal Topic Resources Standards Chapter 4 Sections 2, 5-6 p , CC.2.2.HS.C.5 Chapter 3 Section 5 p Greatest Integer Chapter 2 Section 6 p /Calculus Page 7

9 Introduction to Derivatives and Integrals Dunmore School District resources EK1.1A1 EK1.1B1 EK1.1C3 EK2.1A1 EK2.1A2 EK2.1A3 EK2.1A4 EK2.1C1 EK2.1C2 EK2.1C3 EK2.1C4 EK2.1C5 EK3.1A1 EK3.1A2 EK3.2A1 EK3.2A2 EK3.2C1 EK3.2C2 EK3.2C3 Final Review and Exam /Calculus Page 8

10 General Topic Anchor Descriptor PA Core Standards Eligible Content, Essential Knowledge, Skills & Vocabulary CC.2.3.HS.A.7 Introduction to Trigonometric with Right Apply trigonometric ratios to solve Ratios Triangles problems involving right triangles G Use the Pythagorean theorem to write and/or solve problems involving right triangles G Use trigonometric ratios to write and/or solve problems involving right triangles. HSG.SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Solving for a Side Solving for an Angle Solving the Triangle Application Problems Complementary Angles Reciprocal Ratios Special Right Triangles and Trigonometric Ratios Resources & Activities Chapter 5 Sections 2-4 p Assessments Suggested Time (In Days) 14 days HSG.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. HSG.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. /Calculus Page 9

11 Unit Circle Definition of Trigonometric HSF.TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. HSF.TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Radians Unit Circle Definitions Trigonometric Values of Special Angles Quadrant Angles Reference Angles Chapter 6 Sections 1-2 p days HSF.TF.A.3 Use special right triangles to determine 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 x, π + x, and 2π x in terms of their values for x, where x is any real number. CC.2.2.HS.C.7 Apply radian measure of an angle and the unit circle to analyze the trigonometric functions. /Calculus Page 10

12 Graphs of Trigonometric HSF.TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. HSF.TF.B.5 Choose trigonometric functions to model periodic phenomena with specific amplitude, frequency, and midline. CC.2.2.HS.C.8 Choose trigonometric functions to model periodic phenomena and describe the properties of the graphs. Graphs of sine, cosine and tangent Features of Sinusoidal from their Graphs Finding the Amplitude and Midline from their Formulas Periods of Sinusoidal Graphing Sinusoidal Constructing Sinusoidal Real World Problems Chapter 6 Sections 3-6 p days Trigonometric Algebra CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. G Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction). G Use the Pythagorean Theorem to write and/or solve problems involving right triangles. Fundamental Identities Proving and Verifying Identities Sum and Difference Identities Double-Angle Identities Half-Angle Identities Solving Trigonometric Equations Chapter 7 Sections 1-4, 6-7 p , days /Calculus Page 11

13 G Use trigonometric ratios to write and/or solve problems involving right triangles. HSF.TF.C.8 Prove the Pythagorean identity sin 2 θ + cos 2 θ = 1 and use it to find sin θ, cos θ, or tan θ given sin θ, cos θ, or tan θ and the quadrant of the angle. with General Triangles HSF.TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. HSG.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems. HSG.SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. HSG.SRT.D.9 Derive the formula A=1/2absinC for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Law of Sines Law of Cosines Solving Triangles Area Application Problems Chapter 8 Sections 1-2 p days /Calculus Page 12

14 Introduction to CC C.1 Define, evaluate, and compare functions. Interpret the effects transformations have on functions and find the inverses of functions. Definition Function Notation Evaluating Domain and Range Inverses Parent Function Graphs and Transformations Chapter 2 Section 3 p Chapter 4 Section 1 p days Linear Use the concept and notation of functions to interpret and apply them in terms of their context. Solving Linear Equations Solving Systems Of Equations Word Problems Linear Regression Chapter 2 Section 4 p days Graph and analyze functions and use their properties to make connections between the different representations. Graphing Linear Analyzing the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness /Calculus Page 13

15 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. Interpret functions in terms of the situations they model. Local extrema Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Polynomial CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Write functions or sequences that model relationships between two quantities. Quadratic Solving Graphing Application Problems Cubic Solving Graphing Application Problems Analyzing the Domain/Range Continuity Increasing/decreasing behav ior Symmetry Boundedness Local extrema Horizontal asymptotes Chapter 3 Sections 1, 3-4 p , days /Calculus Page 14

16 Interpret the effects transformations have on functions and find the inverses of functions CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. Interpret functions in terms of the situations they model. Vertical asymptotes End behavior x-intercepts y-intercepts Absolute Value Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Write functions or sequences that model relationships between two quantities. Interpret the effects transformations have on functions and find the inverses of functions Solving Absolute Value Equations Graphing Absolute Value Analyze the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness Local extrema Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Chapter 1 Section 8 p days /Calculus Page 15

17 Interpret functions in terms of the situations they model. Square Root Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Write functions or sequences that model relationships between two quantities. Solving Radical Equations Graphing Square Root Analyze the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness Local extrema Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Review Chapter Section 7 p Chapter 1 Section 6 p Chapter 2 Section 6 p days Interpret the effects transformations have on functions and find the inverses of functions Interpret functions in terms of the situations they model. /Calculus Page 16

18 Logarithmic Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Write functions or sequences that model relationships between two quantities. Interpret the effects transformations have on functions and find the inverses of functions Solving Logarithmic Graphing Logarithmic Analyze the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness Local extrema Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Chapter 4 Sections 3-5 p days Interpret functions in terms of the situations they model. /Calculus Page 17

19 Exponential Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Write functions or sequences that model relationships between two quantities. Interpret the effects transformations have on functions and find the inverses of functions Solving Exponential Equations Growth and Decay Interest Problems Graphing Exponential Analyze the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness Local extrema Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Chapter 4 Sections 2, 5-6 p , days CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. Interpret functions in terms of the situations they model. /Calculus Page 18

20 Reciprocal Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Write functions or sequences that model relationships between two quantities.. Interpret functions in terms of the situations they model. Solving Reciprocal Graphing Reciprocal Analyze the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness Local extrema Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Chapter 3 Section 5 p days Greatest Integer Use the concept and notation of functions to interpret and apply them in terms of their context. Graph and analyze functions and use their properties to make connections between the different representations. Evaluating Greatest Integer Graphing Greatest Integer Analyze the Function Domain/Range Continuity Increasing/decreasing behavior Symmetry Boundedness Local extrema Chapter 2 Section 6 p days /Calculus Page 19

21 Write functions or sequences that model relationships between two quantities. Interpret the effects transformations have on functions and find the inverses of functions Interpret functions in terms of the situations they model. Horizontal asymptotes Vertical asymptotes End behavior x-intercepts y-intercepts Introduction to Derivatives and Integrals EK1.1A1 Given A function f, the limit f(x) as x approaches c is a real number R f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is lim x c f(x) = R. EK1.1B1 Numerical and graphical information can be used to estimate limits. Limits Power Rule for Derivatives Power Rule for Integrals Applications Area under the curve Displacement, velocity, acceleration resources. 15 days EK1.1C3 Limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules. /Calculus Page 20

22 EK2.1A1 The difference quotients f(a+h) f(a) h and f(x) f(a) express the average rate x a of change of a function over an interval. EK2.1A2 The instantaneous rate of change of a function at a point can be expressed by lim f(a+h) f(a) h 0 h or lim f(x) f(a) x a x a provided that the limit exists. These are common forms of the definition of the derivative and are denoted f (a). EK2.1A3 The derivative of f is the functions whose value at x is lim provided this limit exists. f(a+h) f(a) h 0 h EK2.1A4 For y=f(x), notations for the derivative include dy dx, f (x), and y. EK2.1C1 Direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic functions., /Calculus Page 21

23 EK2.1C2 Specific rules can be used to calculate derivatives for classes for functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric. EK2.1C3 Sums, differences, products, and quotients of functions can be differentiated using derivative rules. EK2.1C4 The chain rule provides a way to differentiate composite functions. EK2.1C5 The chain rule is the basis for implicit differentiation. EK3.1A1 An antiderivative of a function f is a function g whose derivative is f. EK3.1A2 Differentiation rules provide the foundation for finding antiderivatives. EK3.2A1 A Riemann sum, which requires a partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval /Calculus Page 22

24 multiplied by the length of that subinterval of the partition. EK3.2A2 The definite integral of a continuous function f over the interval [a, b], b denoted by f(x)dx = a n lim f(x i) x i where x i is a i=1 max x i 0 value in the ith subinterval, Δx i is the width of the ith subinterval, n is the number of subintervals, and maxδx i is the width of the largest subinterval. Another form of the definition is b a f(x)dx = lim n i=1 n where x i = b a n the ith subinterval. f(x i ) x i, and xi is the value in EK3.2C1 In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area. EK3.2C2 Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals. /Calculus Page 23

25 Final Review EK3.2C3 The definition of the definite integral may be extended to functions with removable or jump discontinuities. 16 days /Calculus Page 24

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