Grade 7 Curriculum Guide s2

2 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS This page is intentionally left blank. TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Introduction

The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn.

The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, Teacher Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below.

Curriculum Information: This section includes the objective, focus or topic, and in some, not all, foundational objectives that are being built upon.

Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective.

Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.

Essential Questions and Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives.

Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning.

Resources: This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.

Sample Instructional Strategies and Activities: This section lists ideas and suggestions that teachers may use when planning instruction. 4 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS

The following chart lists the objectives for the Prince William County Trigonometry Curriculum. The chart organizes the objectives by topic. The Prince William County cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.

Topic Objectives Triangular and Circular Trigonometric Functions T1, T2, T3 Inverse Trigonometric Functions T4, T7 Trigonometric Identities T5 Trigonometric Equations, Graphs, and Practical Problems T6, T8, T9

5 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions Triangular and Circular Trigonometric mathematical communication,  What is the standard position of an angle? Functions mathematical reasoning, connections  Given a point on the terminal side of an angle, how are the values of the six and representations to: trigonometric functions determined? Virginia Standard T.1  Define the six triangular trigonometric What is the relationship between trigonometric and circular functions? The student, given a point other than functions of an angle in a right triangle.  the origin on the terminal side of the  Define the six circular trigonometric angle, will use the definitions of the six functions of an angle in standard Essential Understandings trigonometric functions to find the sine, position.  Triangular trigonometric function definitions are related to circular trigonometric cosine, tangent, cotangent, secant, and  Make the connection between the function definitions. cosecant of the angle in standard triangular and circular trigonometric  Both degrees and radians are units for measuring angles. position. Trigonometric functions functions.  Drawing an angle in standard position will force the terminal side to lie in a specific defined on the unit circle will be related  Recognize and draw an angle in quadrant. to trigonometric functions defined in standard position.  A point on the terminal side of an angle determines a reference triangle from which the right triangles.  Show how a point on the terminal side values of the six trigonometric functions may be derived. of an angle determines a reference triangle. Teacher Notes and Elaborations As derived from the Greek language, the word trigonometry means “measurement of triangles”. Key Vocabulary circular trigonometric function An angle is determined by rotating a ray (half-line) about its endpoint. The starting position degrees of the ray is the initial side of the angle, and the position after rotation is the terminal side. initial side radians The six trigonometric functions of an angle θ are called sine, cosine, tangent, cotangent, reference triangle secant and cosecant. The functions are defined with the angle θ (the Greek letter theta) in terminal side standard position. triangular trigonometric function unit circle In the rectangular coordinate system an angle with its vertex at the origin and with its initial side along the positive x-axis is in standard position. For any point P(x, y) on the terminal side of an angle θ in standard position, r is defined as the distance from the vertex to P . A point on the terminal side of an angle determines a reference triangle from which the values of the six trigonometric functions may be derived.

6 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS (continued)

Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations

7 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic Teacher Notes and Elaborations (continued) Triangular and Circular Trigonometric The six triangular trigonometric functions of θ are: Functions

Virginia Standard T.1 The student, given a point other than the origin on the terminal side of the The properties of the trigonometric functions are connected with the circular function definitions by using a unit circle (a circle with the angle, will use the definitions of the six radius of one). trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and If the terminal side of an angle θ in standard position intersects the unit circle at P(x, y), then the six circular trigonometric functions are cosecant of the angle in standard defined as: position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. Degrees and radians are equivalent units for angle measurement. One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.

8 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Resources Sample Instructional Strategies and Activities

Topic Text: Triangular and Circular Trigonometric Trigonometry, Sixth Edition, 2006, Functions McDougal Littell/Houghten Mifflin Virginia Standard T.1 PWC Mathematics website http://pwcs.math.schoolfusion.us/

Virginia Department of Education website http://www.doe.virginia.gov/instruction/ma thematics/index.shtml

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Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions Triangular and Circular Trigonometric mathematical communication,  What are the Pythagorean, ratio, and reciprocal identities? Functions mathematical reasoning, connections  Given the value of one trigonometric function, how are the remaining functions and representations to: determined?  Given one trigonometric function value, Virginia Standard T.2 find the other five trigonometric Essential Understandings The student, given the value of one function values.  If one trigonometric function value is known, then a triangle can be formed to use in trigonometric function, will find the  Develop the unit circle, using both finding the other five trigonometric function values. values of the other trigonometric degrees and radians.  Knowledge of the unit circle is a useful tool for finding all six trigonometric values for functions, using the definitions and  Solve problems, using the circular special angles. properties of the trigonometric function definitions and the properties functions. of the unit circle. Teacher Notes and Elaborations  Recognize the connections between the Given the value of one trigonometric function, a triangle can be formed to use in finding the coordinates of points on a unit circle other five trigonometric function values or the remaining functions may also be found using and one of the following methods: - coordinate geometry; Definitions of the trigonometric functions: - cosine and sine values; and and the - lengths of sides of special right and the triangles (30° - 60° - 90° and and the 45° - 45° - 90°). Relationships between trigonometric functions are identities. Reciprocal Identities: Key Vocabulary Since and the , then and . degrees Also, cos θ and sec θ are reciprocals as are tan θ and cot θ. The reciprocal identities Pythagorean identities hold for any angle θ that does not lead to a zero denominator. radians ratio (quotient) identities Pythagorean Identities: reciprocal identities unit circle

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11 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Triangular and Circular Trigonometric Ratio or Quotient Identities: Functions

Virginia Standard T.2 Degrees and radians are equivalent units for angle measurement. A central angle with sides and intercepted arcs all the same length The student, given the value of one measures 1 radian. trigonometric function, will find the values of the other trigonometric A unit circle is one that lies on the x-axis, has origin (0, 0), and a radius of 1. functions, using the definitions and properties of the trigonometric Knowledge of the unit circle is a useful tool for finding all six trigonometric values for special angles. functions.

Curriculum Information Resources Sample Instructional Strategies and Activities

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Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations

15 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions Triangular and Circular Trigonometric mathematical communication,  What is the relationship between radians and degrees? Functions mathematical reasoning, connections  What is the relationship between families of coterminal angles? and representations to:  What is meant by the special angles?  Find trigonometric function values of Virginia Standard T.3 specials angles and their related angles Essential Understandings The student will find, without the aid of in both degrees and radians.  Special angles are widely used in mathematics. a calculator, the values of the  Apply the properties of the unit circle  Unit circle properties will allow special-angle and related-angle trigonometric values to trigonometric functions of the special without using a calculator. be found without the aid of a calculator. angles and their related angles as found  Use a conversion factor to convert from  Degrees and radians are units of angle measure. in the unit circle. This will include radians to degrees and vice versa  A radian is the measure of the central angle that is determined by an arc whose length is converting angle measures from radians without using a calculator. the same as the radius of the circle. to degrees and vice versa. Teacher Notes and Elaborations Key Vocabulary The two most common units used to measure angles are radians and degrees. The radian coterminal angles measure of an angle in standard position is defined as the length of the corresponding arc degrees divided by the radius of the circle (). One degree, 1°, is the result from a rotation of of a quadrantal angles complete revolution about the vertex in the positive direction. A full revolution radian (counterclockwise) corresponds to 360º. revolution unit circle To convert radians to degrees and vice versa, multiply by the appropriate conversion factor .

Multiples, between 0 and 2π, of first quadrant special angles are found without the aid of a calculator.

Angles that measure greater than 2π can be formed by adding or subtracting a multiple of 2π to its coterminal angle measuring between 0 and 2π.

Two angles in standard position with the same initial and terminal sides are called coterminal angles.

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations 16 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic Teacher Notes and Elaborations (continued) Triangular and Circular Trigonometric Special angles are widely used in mathematics. The first quadrant special angles of a unit circle (a circle with a radius of one) are , , . The Functions quadrantal angles (any angle with the terminal side on the x-axis or y-axis) of a unit circle are 0, , π, , 2π.

Virginia Standard T.3 The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.

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Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions Inverse Trigonometric Functions mathematical communication,  What are inverse trigonometric functions? mathematical reasoning, connections and representations to: Essential Understandings Virginia Standard T.4  Use a calculator to find the  The trigonometric function values of any angle can be found by using a calculator. The student will find, with the aid of a trigonometric function values of any  The inverse trigonometric functions can be used to find angle measures whose calculator, the value of any angle in either degrees or radians. trigonometric function values are known. trigonometric function and inverse  Define inverse trigonometric functions.  Calculations of inverse trigonometric function values can be related to the triangular trigonometric function. Find angle measures by using the definitions of the trigonometric functions. inverse trigonometric functions when the trigonometric function values are Teacher Notes and Elaborations given. The values of the trigonometric functions of any angle can be approximated using a calculator. Most values are approximated to four decimal places. Depending upon the problem, calculators must be in the appropriate mode, whether radian or degree. Key Vocabulary inverse trigonometric functions The inverse trigonometric functions can be used to find angle measures whose trigonometric function values are known. Given the value of any trigonometric function, the angle may be determined by using the appropriate inverse function key on the calculator. Values of inverse trigonometric functions are always in radians.

Definitions of the Inverse Trigonometric Functions: Function Domain Range

if and only if sin y = x

if and only if cos y = x

if and only if tan y = x

21 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic Text: Inverse Trigonometric Functions Trigonometry, Sixth Edition, 2006, McDougal Littell/Houghten Mifflin Virginia Standard T.4

PWC Mathematics website http://pwcs.math.schoolfusion.us/

22 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions Trigonometric Identities mathematical communication,  What is an identity? mathematical reasoning, connections  What is the difference between solving equations and verifying identities? and representations to: Virginia Standard T.5  Use trigonometric identities to make Essential Understandings The student will verify basic algebraic substitutions to simplify and trigonometric identities and make verify trigonometric identities. The  Trigonometric identities can be used to simplify trigonometric expressions, equations, or substitutions, using the basic identities. basic trigonometric identities include identities. - reciprocal identities;  Trigonometric identity substitutions can help solve trigonometric equations, verify - Pythagorean identities; another identity, or simplify trigonometric expressions. - sum and difference identities; Teacher Notes and Elaborations - double-angle identities; and An identity is an equation that is true for all possible replacements of the variables. An - half-angle identities. identity involving trigonometric expressions is a trigonometric identity. Trigonometric identities can be used to simplify trigonometric expressions, equations, or identities. The fundamental trigonometric identities are the following: Key Vocabulary - reciprocal identities, identity - Pythagorean identities, double-angle identities half-angle identities - sum and difference identities, Pythagorean identities - half angle identities, and reciprocal identities - double angle identities. sum and difference identities trigonometric identities Reciprocal Identities: verify Since and the , then and . Also, cos θ and sec θ are reciprocals as are tan θ and cot θ. The reciprocal identities hold for any angle θ that does not lead to a zero denominator.

Pythagorean Identities:

Ratio or Quotient Identities:

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23 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Trigonometric Identities Double-Angle Identities:

= Virginia Standard T.5 = The student will verify basic trigonometric identities and make Sum and Difference Identities: substitutions, using the basic identities.

Half-Angle Identities:

The signs of depend on the quadrant in which lies.

To verify a trigonometric identity, either the left or the right side of the equation may be used to deduce the other side. Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same terms to both sides, are not valid when working with identities since the statement to be verified may not be true. To verify an identity, show that one side of the identity can be simplified so that it is identical to the other side.

Guidelines for Verifying Trigonometric Identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even making an attempt that leads to a dead end provides insight. 6. Try working backwards from the solution, as it can provide great insight.

Topic Text: Trigonometric Identities Trigonometry, Sixth Edition, 2006, McDougal Littell/Houghten Mifflin Virginia Standard T.5

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations

27 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions Trigonometric Equations, Graphs, and mathematical communication,  What effect does the change in the values A, B, C, and D in the equation Practical Problems mathematical reasoning, connections y = A sin(Bx - C) + D, have on the graph of the function? and representations to:  Why are the terms: phase shift, period, amplitude, vertical shift and asymptote  Determine the amplitude, period, phase important to curve sketching? Virginia Standard T.6 shift, and vertical shift of a The student, given one of the six trigonometric function from the Essential Understandings trigonometric functions in standard equation of the function and from the form, will  The domain and range of a trigonometric function determine the scales of the axes for graph of the function. the graph of the trigonometric function. a. state the domain and the range of  Describe the effect of changing A, B, C,  The amplitude, period, phase shift, and vertical shift are important characteristics of the the function; or D in the standard form of a graph of a trigonometric function, and each has a specific purpose in applications using b. determine the amplitude, period, trigonometric equation trigonometric equations. phase shift, vertical shift; and {e.g., y = A sin(Bx + C) + D, or  The graph of a trigonometric function can be used to display information about the asymptotes; y = A cos[B(x + C)] + D}. periodic behavior of a real-world situation, such as wave motion or the motion of a c. sketch the graph of the function by  State the domain and the range of a Ferris wheel. using transformations for at least a function written in standard form two-period interval; and {e.g., y = A sin(Bx + C) + D or Teacher Notes and Elaborations d. investigate the effect of changing y = A cos[B(x + C)] + D}. Each of the six trigonometric functions is a periodic function whose graph is based on the parameters in a trigonometric  Sketch the graph of a function written repetition. A periodic function is a function such that for every real number in the domain function on the graph of the in standard form of and for some positive real number. The smallest possible positive value of is the period function. {e.g., y = A sin(Bx + C) + D or of the function. The period of the sine, cosine, secant, and cosecant function is 2π. The y = A cos[B(x + C)] + D } by using period of the tangent and cotangent function is π. transformations for at least a two period interval. The amplitude of a function can be interpreted as half the difference between its maximum and minimum values. The amplitude is half the range (difference between maximum and minimum values). Key Vocabulary amplitude Suggested five steps to sketch the parent graph of y = A sin Bx or y = A cos Bx, with are: asymptote 1. Determine the period of repeat, . Start at 0 on the x-axis and mark off that distance. horizontal phase shift 2. Divide the interval into four equivalent parts. period of the function 3. Evaluate the function for each of the five x values resulting from Step 2. The points periodic function will be maximum points, minimum points, and x intercepts. range 4. Plot those points found in Step 3 and join them with a curve. vertical phase shift 5. Draw additional cycles to the left and right of the curve.

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28 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Trigonometric Equations, Graphs, and Transformations to the original graph can be done through phase shifts. The vertical phase shift moves the horizontal axis of the graph Practical Problems along the y-axis. The horizontal phase shift moves the graph along the x-axis.

Steps to sketch the graph of y = A sin(Bx – C) + D or y = A cos(Bx – C) + D, with are: Virginia Standard T.6 1. Determine D the vertical phase shift. This will be the new horizontal axis at y = D. The student, given one of the six 2. Determine C the horizontal phase shift. This will lie on the x-axis. trigonometric functions in standard Follow steps 1 - 5 above. form, will 3. a. state the domain and the range of The asymptote is a straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases the function; without limit. b. determine the amplitude, period, phase shift, vertical shift; and Reciprocal identities are used to obtain the graphs of the secant and cosecant functions. The cosecant and secant functions will have asymptotes; vertical asymptotes. The asymptotes will have equations of the form , where k is the x-intercept of the sine or cosine function. c. sketch the graph of the function by using transformations for at least a Sketching the graphs of the variations of the tangent and cotangent is similar to sketching the graphs of the transformations of sine and two-period interval; and cosine functions. Key differences are the period of repeat, asymptotes, and the shape of the graph. Tangent and cotangent graphs do not d. investigate the effect of changing have amplitude. the parameters in a trigonometric function on the graph of the The graphing calculator can provide a visual look at how the constants A, B, C, and D affect the graph of a function. Be sure the calculator function. is set for radians. Most calculators have a trig window with domain [-2π, 2π], range [-4, 4], π, and . Other settings may be preferable for different equations.

Graphs of trigonometric functions model periodic behavior of real world situations such as wave motion, biorhythms, seasonal temperatures, or the motion of a Ferris wheel.

30 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic Text: Trigonometric Equations, Graphs, and Trigonometry, Sixth Edition, 2006, Practical Problems McDougal Littell/Houghten Mifflin Virginia Standard T.6 PWC Mathematics website http://pwcs.math.schoolfusion.us/

32 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions and Understandings Inverse Trigonometric Functions mathematical communication,  What are the domains and ranges of the inverse trigonometric functions? mathematical reasoning, connections  What are the restrictions on the domain of the inverse trigonometric functions? Virginia Standard T.7 and representations to: The student will identify the domain  Find the domain and range of the Essential Understandings and range of the inverse trigonometric inverse trigonometric functions.  Restrictions on the domains of some inverse trigonometric functions exist. functions and recognize the graphs of  Use the restrictions on the domains of these functions. Restrictions on the the inverse trigonometric functions in Teacher Notes and Elaborations domains of the inverse trigonometric finding the values of the inverse The trigonometric functions are not one-to-one, so it is necessary to determine the functions will be included. trigonometric functions. restrictions on domains to regions that pass the horizontal line test. The inverse  Identify the graphs of the inverse trigonometric functions can be denoted in two ways. For example, the inverse of may be trigonometric functions. written as or .

Function Domain Range Key Vocabulary y = arcsin x [-1,1] inverse trigonometric function y = arccos x [-1,1] [0, π] restrictions on domains y = arctan x [-∞,∞] y = arccot x [-∞,∞] [0, π]

Function Domain y = arcsec x [-∞, -1][1, ∞] y = arccsc x [-∞, -1][1, ∞]

Function Range y = arcsec x [0,π], y = arccsc x ,

The graphs of the inverse trigonometric functions are obtained by interchanging the x- and y- coordinates of the key points of the basic graphs.

Topic Text: Inverse Trigonometric Functions Trigonometry, Sixth Edition, 2006, McDougal Littell/Houghten Mifflin Virginia Standard T.7

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions and Understandings Trigonometric Equations, Graphs, and mathematical communication,  Do trigonometric equations have unique solutions? Why or why not? Practical Problems mathematical reasoning, connections  What is the relationship of the domain and range to the solution of trigonometric and representations to: equations?  Solve trigonometric equations with Virginia Standard T.8 restricted domains algebraically and by Essential Understandings The student will solve trigonometric using a graphing utility.  Solutions for trigonometric equations will depend on the domains. equations that include both infinite  Solve trigonometric equations with solutions and restricted domain infinite solutions algebraically and by  A calculator can be used to find the solution of a trigonometric equation as the points of solutions and solve basic trigonometric using a graphing utility. intersection of the graphs when one side of the equation is entered in the calculator as Y1 inequalities.  Check for reasonableness of results, and and the other side is entered as Y2. verify algebraic solutions, using a graphing utility. Teacher Notes and Elaborations Trigonometric equations, like most algebraic equations, are true for some, but not for all values of the variable. Trigonometric equations do not have unique solutions. Solutions for Key Vocabulary trigonometric equations will depend on the domains. They have infinitely many solutions, trigonometric equation differing by the period of the function. If the domain of the equations is restricted to one trigonometric identities revolution then only those solutions between 0 and 2π will be determined. To solve a trigonometric equation, use standard algebraic techniques and fundamental trigonometric identities.

The fundamental trigonometric identities are the following: - reciprocal identities, - Pythagorean identities, - sum and difference identities, - half angle identities, and - double angle identities.

Standard algebraic techniques are used to solve trigonometric inequalities.

36 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Topic Text: Trigonometric Equations, Graphs, and Trigonometry, Sixth Edition, 2006, Practical Problems McDougal Littell/Houghten Mifflin Virginia Standard T.8 PWC Mathematics website http://pwcs.math.schoolfusion.us/

37 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions and Understandings Trigonometric Equations, Graphs, and mathematical communication,  How are practical problems involving triangles and vectors solved? Practical Problems mathematical reasoning, connections  What is the relationship of a vector to right triangles and trigonometric functions? and representations to: What is meant by an ambiguous case when determining parts of a triangle? Virginia Standard T.9  Write a real-world problem involving  The student will identify, create, and triangles. solve real-world problems involving  Solve real-world problems involving Essential Understandings triangles. Techniques will include using triangles.  A real-world problem may be solved by using one of a variety of techniques associated the trigonometric functions, the  Use the trigonometric functions, with triangles. Pythagorean Theorem, the Law of Pythagorean Theorem, Law of Sines, Sines, and the Law of Cosines. and Law of Cosines to solve real-world Teacher Notes and Elaborations problems. Practical problems involving right triangles can be solved by applying the right triangle  Use the trigonometric functions to definitions of trigonometric functions and the Pythagorean Theorem. Problems involving model real-world situations. oblique (non-right) triangles are solved using the Law of Sines or the Law of Cosines  Identify a solution technique that could depending upon the given information. be used with a given problem.  Prove the addition and subtraction The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides formulas for sine, cosine, and tangent of lengths a, b, and c (a opposite , , and ). and use them to solve problems. The Law of Cosines states that for any triangle with sides of lengths a, b, and c then . Key Vocabulary directed line segment To solve an oblique triangle, the measure of at least one side and any two other parts of the Law of Cosines triangle need to be known. This breaks down into the following cases. Law of Sines magnitudes Given oblique AAS Law of Sines Pythagorean Theorem ASA Law of Sines scalar SSA Law of Sines (ambiguous case) sum and difference formulas SAS Law of Cosines vector SSS Law of Cosines vector quantity Heron’s area formula is used if the lengths of the sides of the triangle are known. If two sides of a triangle and the angle between the two sides are known then the area formula below is used:

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38 TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Trigonometric Equations, Graphs, and Many quantities in mathematics involve magnitudes. These quantities are called scalar. Other quantities called vector quantities, involve Practical Problems both magnitude and direction. A vector quantity is often represented with a directed line segment, which is called a vector. The length of the vector represents the magnitude of the vector quantity. Each vector has a horizontal and vertical component. Vectors may be added and Virginia Standard T.9 subtracted. The student will identify, create, and solve real-world problems involving Sum and Difference Formulas: triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

Topic Text: Trigonometric Equations, Graphs, and Trigonometry, Sixth Edition, 2006, Practical Problems McDougal Littell/Houghten Mifflin Virginia Standard T.9 PWC Mathematics website http://pwcs.math.schoolfusion.us/