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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS

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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSIntroduction

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.

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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 ObjectivesTriangular and Circular Trigonometric Functions T1, T2, T3Inverse Trigonometric Functions T4, T7Trigonometric Identities T5Trigonometric Equations, Graphs, and Practical Problems T6, T8, T9

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Curriculum Information Essential Knowledge and SkillsKey Vocabulary

Essential Questions and UnderstandingsTeacher Notes and Elaborations

Topic Triangular and Circular Trigonometric Functions

Virginia Standard T.1 The student, given a point other than the origin on the terminal side of the angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Define the six triangular trigonometric

functions of an angle in a right triangle. Define the six circular trigonometric

functions of an angle in standard position.

Make the connection between the triangular and circular trigonometric functions.

Recognize and draw an angle in standard position.

Show how a point on the terminal side of an angle determines a reference triangle.

Key Vocabularycircular trigonometric functiondegreesinitial sideradiansreference triangleterminal sidetriangular trigonometric functionunit circle

Essential Questions What is the standard position of an angle? Given a point on the terminal side of an angle, how are the values of the six

trigonometric functions determined? What is the relationship between trigonometric and circular functions?

Essential Understandings Triangular trigonometric function definitions are related to circular trigonometric

function definitions. Both degrees and radians are units for measuring angles. Drawing an angle in standard position will force the terminal side to lie in a specific

quadrant. 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.

Teacher Notes and ElaborationsAs derived from the Greek language, the word trigonometry means “measurement of triangles”.

An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side.

The six trigonometric functions of an angle θ are called sine, cosine, tangent, cotangent, secant and cosecant. The functions are defined with the angle θ (the Greek letter theta) in standard position.

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.

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Curriculum Information Essential Questions and UnderstandingsTeacher Notes and Elaborations

Topic Triangular and Circular Trigonometric Functions

Virginia Standard T.1 The student, given a point other than the origin on the terminal side of the angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

Teacher Notes and Elaborations (continued)The six triangular trigonometric functions of θ are:

The properties of the trigonometric functions are connected with the circular function definitions by using a unit circle (a circle with the radius of one).

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 defined as:

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.

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Curriculum Information Resources Sample Instructional Strategies and Activities

TopicTriangular and Circular Trigonometric Functions

Virginia Standard T.1

Text:Trigonometry, Sixth Edition, 2006,McDougal Littell/Houghten Mifflin

PWC Mathematics websitehttp://pwcs.math.schoolfusion.us/

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

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http://www.doe.virginia.gov/instruction/mathematics/index.shtml


http://www.pwcsmath.com/


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Virginia Standard T.2 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Given one trigonometric function value,

find the other five trigonometric function values.

Develop the unit circle, using both degrees and radians.

Solve problems, using the circular function definitions and the properties of the unit circle.

Recognize the connections between the coordinates of points on a unit circle and- coordinate geometry;- cosine and sine values; and - lengths of sides of special right

triangles (30° - 60° - 90° and 45° - 45° - 90°).

Key VocabularydegreesPythagorean identitiesradiansratio (quotient) identitiesreciprocal identitiesunit circle

Essential Questions What are the Pythagorean, ratio, and reciprocal identities? Given the value of one trigonometric function, how are the remaining functions

determined?

Essential Understandings If one trigonometric function value is known, then a triangle can be formed to use in

finding the other five trigonometric function values. Knowledge of the unit circle is a useful tool for finding all six trigonometric values for

special angles.

Teacher Notes and ElaborationsGiven the value of one trigonometric function, a triangle can be formed to use in finding the other five trigonometric function values or the remaining functions may also be found using one of the following methods:Definitions of the trigonometric functions:

and the

and the

and the

Relationships between trigonometric functions are identities.Reciprocal Identities:

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:

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Virginia Standard T.2 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions.

Teacher Notes and Elaborations (continued)Ratio or Quotient Identities:

Degrees and radians are equivalent units for angle measurement. A central angle with sides and intercepted arcs all the same length measures 1 radian.

A unit circle is one that lies on the x-axis, has origin (0, 0), and a radius of 1.

Knowledge of the unit circle is a useful tool for finding all six trigonometric values for special angles.

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Text:Trigonometry, Sixth Edition, 2006, McDougal Littell/Houghten Mifflin



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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicTriangular and Circular Trigonometric Functions

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.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find trigonometric function values of

specials angles and their related angles in both degrees and radians.

Apply the properties of the unit circle without using a calculator.

Use a conversion factor to convert from radians to degrees and vice versa without using a calculator.

Key Vocabularycoterminal anglesdegreesquadrantal anglesradianrevolutionunit circle

Essential Questions What is the relationship between radians and degrees? What is the relationship between families of coterminal angles? What is meant by the special angles?

Essential Understandings Special angles are widely used in mathematics. Unit circle properties will allow special-angle and related-angle trigonometric values to

be found without the aid of a calculator. Degrees and radians are units of angle measure. A radian is the measure of the central angle that is determined by an arc whose length is

the same as the radius of the circle.

Teacher Notes and ElaborationsThe two most common units used to measure angles are radians and degrees. The radian measure of an angle in standard position is defined as the length of the corresponding arc

divided by the radius of the circle ( ). One degree, 1°, is the result from a rotation of

of a complete revolution about the vertex in the positive direction. A full revolution (counterclockwise) corresponds to 360º.

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.

(continued)

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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Essential Questions and Understandings

Teacher Notes and ElaborationsTopicTriangular and Circular Trigonometric Functions

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.

Teacher Notes and Elaborations (continued)

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 quadrantal angles (any angle with the terminal side on the x-axis or y-axis) of a unit circle are 0, , π, , 2π.

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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSThis page is intentionally left blank.



TopicInverse Trigonometric Functions

Virginia Standard T.4 The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use a calculator to find the

trigonometric function values of any angle in either degrees or radians.

Define inverse trigonometric functions. Find angle measures by using the

inverse trigonometric functions when the trigonometric function values are given.

Key Vocabularyinverse trigonometric functions

Essential Questions What are inverse trigonometric functions?

Essential Understandings The trigonometric function values of any angle can be found by using a calculator. The inverse trigonometric functions can be used to find angle measures whose

trigonometric function values are known. Calculations of inverse trigonometric function values can be related to the triangular

definitions of the trigonometric functions.

Teacher Notes and ElaborationsThe 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.

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

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TopicTrigonometric Identities

Virginia Standard T.5The student will verify basic trigonometric identities and make substitutions, using the basic identities.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use trigonometric identities to make

algebraic substitutions to simplify and verify trigonometric identities. The basic trigonometric identities include - reciprocal identities; - Pythagorean identities; - sum and difference identities;- double-angle identities; and- half-angle identities.

Key Vocabularyidentitydouble-angle identitieshalf-angle identitiesPythagorean identitiesreciprocal identitiessum and difference identitiestrigonometric identitiesverify

Essential Questions What is an identity? What is the difference between solving equations and verifying identities?

Essential Understandings Trigonometric identities can be used to simplify trigonometric expressions, equations, or

identities. Trigonometric identity substitutions can help solve trigonometric equations, verify

another identity, or simplify trigonometric expressions.

Teacher Notes and ElaborationsAn identity is an equation that is true for all possible replacements of the variables. An 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:

- reciprocal identities,- Pythagorean identities,- sum and difference identities,- half angle identities, and- double angle identities.

Reciprocal Identities:

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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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicTrigonometric Identities

Virginia Standard T.5The student will verify basic trigonometric identities and make substitutions, using the basic identities.

Teacher Notes and Elaborations (continued)

Double-Angle Identities:

= =

Sum and Difference 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 Identities1. 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.

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

TopicTrigonometric Identities





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TopicTrigonometric Equations, Graphs, and Practical Problems

Virginia Standard T.6 The student, given one of the six trigonometric functions in standard form, willa. state the domain and the range of

the function;b. determine the amplitude, period,

phase shift, vertical shift; and asymptotes;

c. sketch the graph of the function by using transformations for at least a two-period interval; and

d. investigate the effect of changing the parameters in a trigonometric function on the graph of the function.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Determine the amplitude, period, phase

shift, and vertical shift of a trigonometric function from the equation of the function and from the graph of the function.

Describe the effect of changing A, B, C, or D in the standard form of a trigonometric equation {e.g., y = A sin(Bx + C) + D, or y = A cos[B(x + C)] + D}.

State the domain and the range of a function written in standard form {e.g., y = A sin(Bx + C) + D ory = A cos[B(x + C)] + D}.

Sketch the graph of a function written in standard form {e.g., y = A sin(Bx + C) + D or y = A cos[B(x + C)] + D } by using transformations for at least a two period interval.

Key Vocabularyamplitudeasymptotehorizontal phase shiftperiod of the functionperiodic functionrangevertical phase shift

Essential Questions What effect does the change in the values A, B, C, and D in the equation

y = A sin(Bx - C) + D, have on the graph of the function? Why are the terms: phase shift, period, amplitude, vertical shift and asymptote

important to curve sketching?

Essential Understandings The domain and range of a trigonometric function determine the scales of the axes for

the graph of the trigonometric function. The amplitude, period, phase shift, and vertical shift are important characteristics of the

graph of a trigonometric function, and each has a specific purpose in applications using trigonometric equations.

The graph of a trigonometric function can be used to display information about the periodic behavior of a real-world situation, such as wave motion or the motion of a Ferris wheel.

Teacher Notes and ElaborationsEach of the six trigonometric functions is a periodic function whose graph is based on

repetition. A periodic function is a function such that for every real

number in the domain of and for some positive real number . The smallest possible positive value of is the period of the function. The period of the sine, cosine, secant, and cosecant function is 2π. The period of the tangent and cotangent function is π.

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).

Suggested five steps to sketch the parent graph of y = A sin Bx or y = A cos Bx, with are:

1. Determine the period of repeat, . Start at 0 on the x-axis and mark off that distance.

2. Divide the interval into four equivalent parts.3. Evaluate the function for each of the five x values resulting from Step 2. The points

will be maximum points, minimum points, and x intercepts.4. Plot those points found in Step 3 and join them with a curve.5. Draw additional cycles to the left and right of the curve.

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



Virginia Standard T.6 The student, given one of the six trigonometric functions in standard form, willa. state the domain and the range of

the function;b. determine the amplitude, period,

phase shift, vertical shift; and asymptotes;

c. sketch the graph of the function by using transformations for at least a two-period interval; and

d. investigate the effect of changing the parameters in a trigonometric function on the graph of the function.

Teacher Notes and Elaborations (continued)Transformations to the original graph can be done through phase shifts. The vertical phase shift moves the horizontal axis of the graph 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:1. Determine D the vertical phase shift. This will be the new horizontal axis at y = D.2. Determine C the horizontal phase shift. This will lie on the x-axis.3. Follow steps 1 - 5 above.

The asymptote is a straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases without limit.

Reciprocal identities are used to obtain the graphs of the secant and cosecant functions. The cosecant and secant functions will have vertical asymptotes. The asymptotes will have equations of the form , where k is the x-intercept of the sine or cosine function.

Sketching the graphs of the variations of the tangent and cotangent is similar to sketching the graphs of the transformations of sine and cosine functions. Key differences are the period of repeat, asymptotes, and the shape of the graph. Tangent and cotangent graphs do not have amplitude.

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

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.

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Virginia Standard T.7The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find the domain and range of the

inverse trigonometric functions. Use the restrictions on the domains of

the inverse trigonometric functions in finding the values of the inverse trigonometric functions.

Identify the graphs of the inverse trigonometric functions.

Key Vocabularyinverse trigonometric functionrestrictions on domains

Essential Questions and Understandings What are the domains and ranges of the inverse trigonometric functions? What are the restrictions on the domain of the inverse trigonometric functions?

Essential Understandings Restrictions on the domains of some inverse trigonometric functions exist.

Teacher Notes and ElaborationsThe trigonometric functions are not one-to-one, so it is necessary to determine the restrictions on domains to regions that pass the horizontal line test. The inverse

trigonometric functions can be denoted in two ways. For example, the inverse of

may be written as or .

Function Domain Range

y = arcsin x [-1,1]y = arccos x [-1,1] [0, π]

y = arctan x [-∞,∞]y = arccot x [-∞,∞] [0, π]

Function Domainy = 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 27

TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSy- coordinates of the key points of the basic graphs.







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TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicTrigonometric Equations, Graphs, and Practical Problems

Virginia Standard T.8 The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Solve trigonometric equations with

restricted domains algebraically and by using a graphing utility.

Solve trigonometric equations with infinite solutions algebraically and by using a graphing utility.

Check for reasonableness of results, and verify algebraic solutions, using a graphing utility.

Key Vocabularytrigonometric equationtrigonometric identities

Essential Questions and Understandings Do trigonometric equations have unique solutions? Why or why not? What is the relationship of the domain and range to the solution of trigonometric

equations?

Essential Understandings Solutions for trigonometric equations will depend on the domains. A calculator can be used to find the solution of a trigonometric equation as the points of

intersection of the graphs when one side of the equation is entered in the calculator as Y1

and the other side is entered as Y2.

Teacher Notes and ElaborationsTrigonometric 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 trigonometric equations will depend on the domains. They have infinitely many solutions, differing by the period of the function. If the domain of the equations is restricted to one 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.

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






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Virginia Standard T.9 The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Write a real-world problem involving

triangles. Solve real-world problems involving

triangles. Use the trigonometric functions,

Pythagorean Theorem, Law of Sines, and Law of Cosines to solve real-world problems.

Use the trigonometric functions to model real-world situations.

Identify a solution technique that could be used with a given problem.

Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Key Vocabularydirected line segmentLaw of CosinesLaw of SinesmagnitudesobliquePythagorean Theoremscalarsum and difference formulasvectorvector quantity

Essential Questions and Understandings How are practical problems involving triangles and vectors solved? What is the relationship of a vector to right triangles and trigonometric functions? What is meant by an ambiguous case when determining parts of a triangle?

Essential Understandings A real-world problem may be solved by using one of a variety of techniques associated

with triangles.

Teacher Notes and ElaborationsPractical problems involving right triangles can be solved by applying the right triangle definitions of trigonometric functions and the Pythagorean Theorem. Problems involving oblique (non-right) triangles are solved using the Law of Sines or the Law of Cosines depending upon the given information.

The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides

of lengths a, b, and c (a opposite , , and ).

The Law of Cosines states that for any triangle with sides of lengths a, b, and c then .

To solve an oblique triangle, the measure of at least one side and any two other parts of the triangle need to be known. This breaks down into the following cases.

GivenAAS Law of SinesASA Law of SinesSSA Law of Sines (ambiguous case)SAS Law of CosinesSSS Law of Cosines

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



Virginia Standard T.9 The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

Teacher Notes and Elaborations (continued)Many quantities in mathematics involve magnitudes. These quantities are called scalar. Other quantities called vector quantities, involve 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 subtracted.

Sum and Difference Formulas:

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