- Trigonometry for Solving Problems -
This lesson offers a pair of puzzles to enforce the skills of identifying equivalent trigonometric expressions. Additional worksheets enhance students' abilities to appreciate and use trigonometry as a tool in problem solving. This lesson is adapted from an article by Mally Moody, which appeared in the March 1992 edition of
*Mathematics Teacher*. - Trigonometry- This web site has definitions, applets, and much more to help students learn about trigonometry and other connected concepts.
- Maths Online- Law of Sines- This resource contains multiple applets and graphics that can be used by students or teachers to understand and visualize trigonometry in action.
- Trigonometry (Applets and Activities)- This resource has multiple applets and activities to be used by the students or teacher for discovery, practice, or review of some basic trigonometric concepts such as definition of sine and cosine, graph of sine and cosine, law of sines, law of cosines, and more.
- The Math Page: Trigonometry- This resource has multiple concepts for geometry and trigonometry. The concepts are divided among chapters with links on common unknown concepts to help students understand the text. This resource also provides exercises that can be done by the students (answers are provided).
- Web Math - Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics. The site also offers specific sections on the conversion of units (applicable to the sciences).

TitleTrigonometry for Solving Problems URLhttp://illuminations.nctm.org/LessonDetail.aspx?id=L383 Materials needed

- Trigonometry Square Activity Sheet 1
- Trigonometry Square Activity Sheet 2
- Activity Sheet 1
- Activity Sheet 2
- Calculators
- Transparencies
Learning ObjectivesStudents will be able to:

- analyze situations, check for limitations, and examine appropriate methods of solutions using trigonometry
- practice manipulating trigonometric functions and in substituting equivalent expressions
- work in small groups encouraging classmates and communicating thoughts
Grade LevelsTrigonometry CA 97 StandardsTrigonometry 1.0 Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between angles and radians. Trigonometry 2.0 Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions. Trigonometry 3.0 Students know the identity cos^2 (x) +sin^2(x)=1. Trigonometry 3.2 Students prove other trigonometric identities and simplify others by using the identity cos^2(x)+sin^2(x)=1. For example, students use this identity to prove that sec^2(x)=tan^2(x)-1. Trigonometry 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. Trigonometry 6.0 Students know the definitions of secant and cosecant functions and can graph them. Trigonometry 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. CA Common Core State StandardsStandards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoningFunctions-TF.1; Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Functions-TF.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. Functions-TF.3; Use special 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, cosines, and tangent for x, π+x, and 2π–x in terms of their values for x, where x is any real number. Functions-TF.8; Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. Functions-TF.9; Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Geometry-SRT.11; Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). How could you use this resource?The teacher can use this activity as a review of trigonometric properties, identities, and concepts. There are puzzles students can work on in a group as well as a worksheet with real world problems. This resource is easy to use. The worksheets are already in a pdf format. EL and Special NeedsEL students will not need to rely heavily on words to complete the puzzle. Students need to know the concepts being addressed in the activity puzzle. The worksheet can be done in a group where certain questions can be explained by others in the group. Groups can also be used for students with special learning needs as well. Other differentiation can be determined by the teacher. Lesson PlansTeacher CommentsCostFree Copyright© 2000-2010 NCTM

TitleTrigonometry URLhttp://mathematica.ludibunda.ch/trigonometry.html Materials needed

- Java software
- Flash 5 or higher
Learning ObjectivesStudents will be able to:

- discover how to use trigonometry applied to multiple concepts
Grade LevelsGeometry, Trigonometry CA 97 StandardsGeometry: 18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. Trigonometry: 2.0 Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle ad are familiar with the graphs of the sine and cosine functions. Trigonometry 4.0 Students graph functions of the form f(t) = A sine(Bt + C) or f(t) = A cos(Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. Trigonometry 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. CA Common Core State StandardsStandards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

5. Use appropriate tools strategically

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoningGeometry-SRT.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. Geometry-SRT.7; Explain and use the relationship between the sine and cosine of complementary angles. Geometry-SRT.8; Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Functions-TF.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. Functions-TF.3; Use special 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, cosines, and tangent for x, π+x, and 2π–x in terms of their values for x, where

x is any real number.Functions-TF.5; Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Functions-TF.8; Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. How could you use this resource?This resource could be used by a student as a research tool. The website has a lot of text with information that could be good if a student was doing a research-based assignment. The applets can be used to help gain further understanding of trigonometry concepts. A teacher can use this resource as a means of preparing a lesson or as a demonstration tool. EL and Special NeedsThis site doesn't address the needs of an EL student very well. However, it does provide definitions which are boldfaced, changed in color, or placed into boxes. The interactive activities on this site can be adopted into a teacher's lesson because they are visible and hands on. Lesson PlansTeacher CommentsCostFree CopyrightNot available on web site

TitleMaths Online- Law of Sines URLhttp://www.univie.ac.at/future.media/moe/galerie/trig/trig.html#dreieck Materials needed

- Java software
- Flash 5 or higher
Learning ObjectivesStudents will be able to:

- discover how to use trigonometry when applied to multiple concepts
Grade LevelsTrigonometry CA 97 StandardsTrigonometry 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. CA Common Core State StandardsStandards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

5. Use appropriate tools strategically

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoningGeometry-SRT.11; Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). How could you use this resource?This resource can be used by students to get extra practice on the law os sines. It can also be used as a discovery tool for students by themselves or as a class with facilitation by the teacher. Teachers can use this resource in a demonstration or monitor students using it in the classroom. EL and Special NeedsVisual applet that lets students experiment with different size triangles to discover the law of sines. Lesson PlansTeacher CommentsCostFree Copyright(c) 1998

TitleTrigonometry (Applets and Activities) URLhttp://www.ies.co.jp/math/java/trig/ Materials neededJava software Learning ObjectivesStudents will be able to:

- discover how to graph trigonometric functions such as sine, cosine, and tangent
- use the law of sines and cosines
- graph trigonometric functions in various forms
Grade LevelsTrigonometry CA 97 StandardsTrigonometry 2.0: Students know the definition of sine and cosine as y- and x- coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions. Trigonometry 4.0: Students graph functions of the form f(t)=Asin(Bt+C) or f(t)=Acos(Bt+C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. Trigonometry 5.0: Students know the definitions of the the tangent and cotangent functions and can graph them. Trigonometry 13.0: Students know the law of sines and the law of cosines and apply those laws to solve problems. CA Common Core State StandardsStandards for Mathematical Practice:

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

5. Use appropriate tools strategically

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoningFunctions-TF.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. Functions-TF.3; Use special 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, cosines, and tangent for x, π+x, and 2π–x in terms of their values for x, where x is any real number. Functions-TF.5; Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Geometry-SRT.11; Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). How could you use this resource?This resource can be used by students to practice trigonometric concepts and develop a better understanding of those concepts through visual representations. The teacher can use this resource as a method of introducing concepts to students through visual representations. EL and Special NeedsEach of these applets has a visual representation of a trigonometric concept that can be explored by the student. Lesson PlansTeacher CommentsCostFree Copyright(c) Not available on web site

TitleThe Math Page: Trigonometry URLhttp://www.themathpage.com/aTrig/trigonometry.htm Materials neededJava software Learning ObjectivesMultiple learning objectives based on subject and subtopic.

Grade LevelsGeometry, Trigonometry CA 97 StandardsMultiple standards for Geometry from the Mathematics Framework for California Public Schools. Multiple standards for Trigonometry from the Mathematics Framework for California Public Schools. CA Common Core State StandardsStandards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoningMultiple standards for Geometry and Trigonometry from the California Common Core State Standards. How could you use this resource?This resource can be used by students as a development tool to better comprehend trigonometric concepts. Teachers can use this resource as a way to find lesson information or to present trigonometric information to students. EL and Special NeedsMany of the explanations for the trigonometric concepts have some color coding to help EL students and special needs students identify parts of the trigonometric concepts. Lesson PlansTeacher CommentsCostFree Copyright© 2001 - 2010 Lawrence Spector

TitleWeb Math URLMaterials neededComputer

Learning ObjectivesLearning objectives vary by concept.

Grade LevelsK-8, Algebra 1, Geometry, Calculus, Trigonometry CA 97 StandardsMultiple standards from the Mathematics Framework for California Public School. CA Common Core State StandardsStandards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoningCovers multiple CaCCSS across grade levels and strands. How could you use this resource?Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics.

The site also offers specific sections on the conversion of units (applicable to the sciences).

EL and Special NeedsVisual graphics display concepts and are interactive. Lesson PlansTeacher CommentsCostFree Copyright(c) 2009 WebMath.com

© 2007 California State University

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