Download Understanding Trigonometric Ratios

Geometry, Quarter 3, Unit 3.3
Understanding Trigonometric Ratios
Overview
Number of instruction days:
9-11
Content to Be Learned
(1 day = 53 minutes)
Mathematical Practices to Be Integrated

Understand definitions of trigonometric ratios
for acute angles in right triangles.
1 Make sense of problems and persevere in
solving them.

Understand that side ratios in right triangles
are properties of the angles in the triangle
because of triangle similarity.



Identify what parts of the triangle need to be
used in order to find a solution.
4 Model with mathematics.
Develop and apply relationships between the
sine and cosine of complementary angles and
use them to solve problems.
Solve right triangles in real-world problems
using trigonometric ratios, special right
triangles, and the Pythagorean Theorem.

Model with mathematics to solve real-world
problems involving trigonometric ratios.
7 Look for and make use of structure.

Recognize the significance of an existing line in
a geometric figure and use the strategy of
drawing an auxiliary line for solving problems.
Step back for an overview.

What is the relationship of the sine and cosine
values of a complementary angle?

How are angles of elevation or depression
similar and different?
Essential Questions

Where would you use each of the following to
solve a problem: trigonometry, special right
triangles, and the Pythagorean Theorem?

Why are trigonometric ratios important?
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Geometry, Quarter 3, Unit 3.3
Understanding Trigonometric Ratios (9-11 days)
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Standards
Common Core State Standards for Mathematical Content
Geometry
Similarity, Right Triangles, and Trigonometry
G-SRT
Define trigonometric ratios and solve problems involving right triangles
G-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.
G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.★
Common Core State Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and
evaluate their progress and change course if necessary. Older students might, depending on the context
of the problem, transform algebraic expressions or change the viewing window on their graphing
calculator to get the information they need. Mathematically proficient students can explain
correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of
important features and relationships, graph data, and search for regularity or trends. Younger students
might rely on using concrete objects or pictures to help conceptualize and solve a problem.
Mathematically proficient students check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?” They can understand the approaches of others to
solving complex problems and identify correspondences between different approaches.
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan
a school event or analyze a problem in the community. By high school, a student might use geometry to
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Understanding Trigonometric Ratios (9-11 days)
Geometry, Quarter 3, Unit 3.3
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solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later.
They are able to identify important quantities in a practical situation and map their relationships using
such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical results in
the context of the situation and reflect on whether the results make sense, possibly improving the
model if it has not served its purpose.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they
may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7
× 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive
property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They
recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an
auxiliary line for solving problems. They also can step back for an overview and shift perspective. They
can see complicated things, such as some algebraic expressions, as single objects or as being composed
of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square
and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
In Grade 4, students classified shapes according to properties of their angles and worked with word
problems involving unknown angle measures. In Grade 5, students classified two-dimensional figures
based on their properties. Students worked with ratios and proportional relationships in Grade 6. A
major cluster for seventh-grade students was extending their ability to recognize, represent, and analyze
proportional relationships. Understanding and applying the Pythagorean Theorem was a major cluster
for Grade 8 students, and they also worked informally to establish facts about angle sums and exterior
angles of triangles.
Current Learning
Fluency with triangle congruency and similarity is expected in Geometry. In this unit, students apply
their knowledge of similar triangles to develop and then apply three trigonometric ratios: sine, cosine,
and tangent. Students add trigonometric ratios to their indirect measurement toolkits, which already
include special right triangles and similar triangles. Students will explore and use the relationship
between sine and cosine ratios of angles whose sum is 90°, the acute angles in a right triangle. Students
understand that by the properties of similarity, side ratios in right triangles are properties of the angles
in the triangle. They solve real-world problems using these relationships. Defining trigonometric ratios
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Geometry, Quarter 3, Unit 3.3
Understanding Trigonometric Ratios (9-11 days)
Version 5
and solving problems involving right triangles is major content as defined by the PARCC Model
Frameworks for Mathematics.
Future Learning
Extensions of the study of trigonometry will include the unit circle and graphing and analyzing
trigonometric functions and identities in Algebra II. This study is further extended in Precalculus to
include solving trigonometric equations. Careers that include use of trigonometry are extensive,
including surveying, engineering, construction, physics, navigation, astronomy, etc.
Additional Findings
This material is challenging for students because they have difficulty distinguishing between the
opposite and adjacent sides in a right triangle. According to Principles and Standards of School
Mathematics, “High school students should develop facility with a broad range of representing
geometric ideas . . .” For example, desks can be arranged in a right triangle to physically represent
change in opposite and adjacent sides when the reference angle is changed. (p. 309)
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your
students’ attainment of the mathematics within the unit.

Develop a definition of trigonometric ratios using corresponding angles of similar right triangles to
show that the relationships of the side ratios are the same.

Apply trigonometric ratios and their inverse relationships to determine missing angle measures and
side lengths of right triangles in problem situations.

Use trigonometric ratios, special right triangles, and the Pythagorean Theorem to solve real world
problems.
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Geometry, Quarter 3, Unit 3.3
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
Solve right triangle problems using angles of elevation and angles of depression.

Apply the relationship between the sine and cosine values of complementary angles to solve
problems.
Instruction
Learning Objectives
Students will be able to:

Use the properties of special right triangles to solve problems.

Compare side and angle measurements in similar right triangles to develop three trigonometric
ratios.

Apply the trigonometric ratios of sine, cosine, and tangent to determine missing side lengths of right
triangles.

Apply inverse relationships and trigonometric ratios to determine missing angle measures in right
triangles from problem situations.

Apply trigonometric ratios, angles of depression, and angles of elevation to solve real-world
problems.

Explore and use the relationship between the sine and cosine values of complementary angles.

Review and demonstrate knowledge of important concepts and procedures related to trigonometric
ratios.
Resources
Geometry, Glencoe McGraw-Hill, 2010, Student/Teacher Editions

Section 8-3 (pp. 552 – 560)

Section 8-4 (pp. 562 – 571)

Section 8-5 (pp. 574 –581)– 493)

http://connected.mcgraw-hill.com/connected/login.do: Glencoe McGraw-Hill Online

Teaching with Manipulatives
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Geometry, Quarter 3, Unit 3.3
Understanding Trigonometric Ratios (9-11 days)
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
Problem Solving Guide (p. 19)

Geometry Lab Transparency Master – Trigonometry (p. 115)

Geometry Lab – Trigonometry (pp. 116 - 117

Chapter 8 Resource Masters (p. 29)

Interactive Classroom CD (PowerPoint Presentations)

Teacher Works CD-ROM
TI-Nspire Teacher Software
Exam View Assessment Suite
Graphic Organizer: SOHCAHTOA. See the Supplementary Unit Materials section of this binder for notes
for this graphic organizer.
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery section below for specific recommendations.
Materials
Ruler, protractor, meter sticks or tape measures (1 for each pair of students), colored pencils, TI-Nspire
graphing calculator, calculator viewscreen, dynamic geometry software; optional – graphic organizer,
clinometers, paper clips, straws, 5 by 7 index cards, and kite string.
Instructional Considerations
Key Vocabulary
angle of depression
sine
angle of elevation
tangent
cosine
trigonometric ratio
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: ratio and special right triangle.
Students who struggle with applications of trigonometric ratios typically do so because they either select
the incorrect ratio or they perform the computations incorrectly. In Section 8-4, to help students
understand how trigonometric ratios relate to similar right triangles, students can elaborate on their
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knowledge by identifying similarities and differences as they compare the ratio of the side lengths and
trigonometric values. This strategy is also helpful as students look for patterns that exist in the
trigonometric values of complementary angles (i.e., sin A = cos B when A + B = 90°).
Refer to the Differentiated Instruction on page 560 and extend this activity to include sine and cosine
trigonometric ratios.
To support students who select incorrect ratios, emphasize the importance of selecting a problemsolving strategy, such as drawing a diagram to help solve the problem. Use colored pencils to color code
the sides of a right triangle. Mnemonic devices such as SOHCAHTOA help students remember the side
lengths involved in each of the three ratios. The four-step problem-solving model and nonlinguistic
representations such as graphic organizers also help struggling students, including English language
learners and students with special needs, organize their knowledge. One resource is Teaching with
Manipulatives on page 19 and another is www.sw-georgia.resa.k12.ga.us/math.html. The SOHCAHTOA
graphic organizer is also provided in the supplementary materials section of this curriculum frameworks
binder.
To help increase students’ computational fluency, provide multiple opportunities for students to
practice using the graphing calculator to compute trigonometric ratios in the context of solving a
problem. Using proportional reasoning to solve equations generated using trigonometric ratios also
helps increase the accuracy of students’ computations, as it connects the idea of solving equations with
trigonometric ratios to the familiar knowledge of solving proportions.
Use real world problem situations to increase the relevance of problems involving angles of elevation
and depression. The Differentiated Instruction for Kinesthetic Learners on page 576 of the Teacher
Edition provides multiple examples of classroom applications. Modeling can also be done with a
clinometer as a tool to help students understand angles of elevation and depression. Numerous
resources on the web reference the integration of clinometers. The following example guides students
through a series of reading and math activities to help them understand how the Northern Lights work,
what causes them, and how to observe them: http://image.gsfc.nasa.gov/poetry/activity/nl4.pdf.
Another opportunity for the integration of a physical models as a nonlinguistic representation is
provided in the Trigonometry Lab, detailed in the Teaching with Manipulatives resource book. In this
activity, students use hypsometers and indirect measurement to calculate measurements of real world
objects.
As you assess students using the 5-minute check transparencies, a cues, questions, and advance
organizers strategy is being used, since students are answering questions about content that is
important. Some of the questions help students review prior knowledge, and these should be used at
the beginning.
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Geometry, Quarter 3, Unit 3.3
Understanding Trigonometric Ratios (9-11 days)
Version 5
Notes
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