# Right Triangles and Trigonometry

Students develop an understanding of right triangles through an introduction to trigonometry, building an appreciation for the similarity of triangles as the basis for developing the Pythagorean theorem.

## Unit Summary

In Unit 4, Right Triangles & Trigonometry, students develop a deep understanding of right triangles through an introduction to trigonometry and the Pythagorean theorem. Students build an appreciation for how similarity of triangles is the basis for developing the Pythagorean theorem and trigonometric properties.

This unit begins with Topic A, Right Triangle Properties and Side-Length Relationships. Students define angle and side-length relationships in right triangles. Students use similarity to prove the Pythagorean theorem and the converse of the Pythagorean theorem. Students develop the algebraic tools to perform operations with radicals. In Topic B, Right Triangle Trigonometry, and Topic C, Applications of Right Triangle Trigonometry, students define trigonometric ratios and make connections to the Pythagorean theorem. Students gain practice with determining an appropriate strategy for solving right triangles. This skill is extended in Topic D, the Unit Circle, where students are introduced to the unit circle and reference angles. From here, students describe how non-right triangles can be solved using the Law of Sines and Law of Cosines, in Topic E. These skills are critical for students' ability to understand calculus and integrals in future years.

There are several lessons in this unit that do not have an explicit common core standard alignment. We have identified that these are important concepts to be introduced in geometry in order for students to access Algebra II and AP Calculus.

Pacing: 21 instructional days (19 lessons, 1 flex day, 1 assessment day)

## Assessment

This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Intellectual Prep

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Internalization of Standards via the Unit Assessment

• Take unit assessment. Annotate for:
• Standards that each question aligns to
• Purpose of each question: spiral, foundational, mastery, developing
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that assessment points to

Internalization of Trajectory of Unit

• Read and annotate "Unit Summary.”
• Notice the progression of concepts through the unit using “Unit at a Glance.”
• Essential understandings
• Connection to assessment questions

### Essential Understandings

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• Trigonometry connects the two features of a triangle—angle measures and side lengths—and provides a set of functions (sine, cosine, tangent), reciprocals, and inverses of those functions to solve triangles given angle measures and side lengths.
• Theorems about right triangles (e.g., Pythagorean theorem, special right triangles, and use of an altitude to make right triangles) give additional tools for finding missing measures.
• Trigonometry, including the Law of Sines, the Law of Cosines, the Pythagorean theorem, trigonometric functions, and inverse trigonometric functions, is used to find measures in real-life applications of inclination, angles of depression, indirect measurement, and various other applications.

### Vocabulary

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 Altitude Leg Hypotenuse Opposite Adjacent Pythagorean theorem Converse of the Pythagorean theorem Radical Radicand Rationalize the denominator Sine Cosine Tangent Trigonometric ratio Arcsine (${\mathrm{sin}^{-1}(\theta)}$) Arccosine (${\mathrm{cos}^{-1}(\theta)}$) Arctangent (${\mathrm{tan}^{-1}(\theta)}$) Angle of elevation Angle of depression Standard position Unit circle Reference angle Area formula for non-right triangles Law of Sines Law of Cosines

### Materials

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• Scientific or graphic calculator

# 1

G.CO.A.1

G.SRT.B.4

Define the parts of a right triangle and describe the properties of an altitude of a right triangle.

# 2

G.SRT.B.4

Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.

# 3

G.SRT.B.4

G.SRT.B.5

Define the relationship between side lengths of special right triangles.

# 4

A.SSE.A.2

N.RN.A.2

Multiply and divide radicals. Rationalize the denominator.

A.SSE.A.2

N.RN.A.2

# 6

G.SRT.C.6

Define and calculate the sine of angles in right triangles. Use similarity criteria to generalize the definition of sine to all angles of the same measure.

# 7

G.SRT.C.6

Define and calculate the cosine of angles in right triangles. Use similarity criteria to generalize the definition of cosine to all angles of the same measure.

# 8

G.SRT.C.7

Derive the relationship between sine and cosine of complementary angles in right triangles, and describe sine and cosine as angle measures approach 0°, 30°, 45°, 60°, and 90°.

# 9

G.SRT.C.6

G.SRT.C.7

Describe and calculate tangent in right triangles. Describe how the value of tangent changes as the angle measure approaches 0°, 45°, and 90°.

# 10

G.SRT.C.8

Solve for missing sides of a right triangle given the length of one side and measure of one angle.

# 11

G.SRT.C.8

Find the angle measure given two sides using inverse trigonometric functions.

# 12

G.SRT.C.8

Describe the relationship between slope and the tangent ratio of the angle of elevation/depression. Use the tangent ratio of the angle of elevation or depression to solve real-world problems.

# 13

G.SRT.C.8

Solve a modeling problem using trigonometry.

# 14

F.TF.A.2

Define angles in standard position and use them to build the first quadrant of the unit circle.

# 15

F.TF.A.2

Use the first quadrant of the unit circle to define sine, cosine, and tangent values outside the first quadrant.

# 16

G.SRT.D.9

Derive the area formula for any triangle in terms of sine.

# 17

G.SRT.D.10

Verify algebraically and find missing measures using the Law of Sines.

# 18

G.SRT.D.10

Verify algebraically and find missing measures using the Law of Cosines.

# 19

G.SRT.D.11

Use side and angle relationships in right and non-right triangles to solve application problems.

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### Congruence
• G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

##### High School — Number and Quantity
• N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

##### Seeing Structure in Expressions
• A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

##### Similarity, Right Triangles, and Trigonometry
• G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

• G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

• G.SRT.C.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.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.

• G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

• G.SRT.D.10 — Prove the Laws of Sines and Cosines and use them to solve problems.

• G.SRT.D.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).

• G.SRT.D.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.

##### Trigonometric Functions
• F.TF.A.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.

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• G.CO.C.10

• 8.EE.A.2

• 8.EE.B.5

• 8.EE.B.6

• 4.G.A.1

• 6.G.A.1

• 8.G.A.1

• 8.G.A.4

• 8.G.B.6

• 8.G.B.7

• 8.G.B.8

• 7.RP.A.2

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• F.TF.A.1

• F.TF.A.2

• F.TF.A.3

• F.TF.A.4

• F.TF.B.5

• F.TF.B.6

• F.TF.B.7

• F.TF.C.8

• F.TF.C.9

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.