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Geometry, Quarter 2, Unit 2.3
Trigonometry
Overview
Number of instructional days:
14
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated

Use appropriate tools strategically.

Use the fact that given an acute angle, all right
triangles that contain the angle must be similar,
to define the tangent, sine, and cosine ratios.
Use the fact that acute angles in right triangles
are complementary to explore useful
relationships between sine and cosine of
complementary angles.

Use technology to explore and deepen
understanding of concepts.
Attend to precision.

Use clear and concise definitions while
communicating reasoning.
Calculate accurately and efficiently.

Find unknown heights, distances, and angle
measures using trigonometric ratios.


Discuss the relationships in special right
triangles. (30–60–90, 45–45–90)
Look for and make use of structure.

Derive a new formula for area of a triangle that
does not depend upon knowing the altitude of
the triangle. A = 1/2 ab sin(C)

Prove the relationships of the Law of Sines and
Law of Cosines.

Use the Law of Sines and Law of Cosines to
find unknown distances and/or angle measures
in real-world problems.

Use the strategy of drawing an auxiliary line
for solving problems.
Essential questions

How do you find the sine, cosine, and tangent
ratios for an acute angle of a right triangle?

How can you use the sine ratio to find a
formula for the area of a triangle?

What can you say about the side lengths and
the trigonometric ratios associated with special
right triangles?

What is the Law of Sines, and how do you use
it to solve problems?


How do you find an unknown angle measure in
a right triangle using trigonometric ratios?
What is the Law of Cosines, and how do you
use it to solve problems?
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-21
Geometry, Quarter 1, Unit 2.3
Trigonometry (14 days)
Written Curriculum
Common Core State Standards for Mathematical Content
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.★
Apply trigonometry to general triangles
G-SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite side.
G-SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
G-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).
Common Core State Standards for Mathematical Practice
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-22
Geometry, Quarter 1, Unit 2.3
Trigonometry (14 days)
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
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 6, students found areas of right triangles and solved ratio problems. In grade 7, students solved
problems with ratio and proportion. Students also used scale drawings to reason about two-dimensional
figures. In grade 8, students learned the Pythagorean Theorem and applied it to find distances between
points on a coordinate plane and to find missing lengths in right triangles. In algebra 1, students solved
quadratic equations.
Current Learning
Students 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. Students explain and use the
relationship between the sine and cosine of complementary angles. Students use trigonometric ratios and
the Pythagorean Theorem to solve right triangles in applied problems. Students derive the formula A = 1/2
ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the
opposite side. Students prove the Law of Sines and the Law of Cosines and use them to solve problems.
Students 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).
Future Learning
Students will use this knowledge in fourth-year courses and college math courses. Students will also use
this knowledge in artwork, engineering, education, constructions, and computer science applications.
Additional Findings
Principles and Standards for School Mathematics states, “Trigonometric relationships are an additional
resource to help students solve math problems” (p. 309).
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-23
Geometry, Quarter 1, Unit 2.3
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
Trigonometry (14 days)
C-24