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.

Assessment

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

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Unit Prep

Intellectual Prep

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.”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Essential Understandings

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

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

Unit Materials, Representations and Tools

Scientific or graphic calculator

Lesson Map

Topic A: Right Triangle Properties and Side-Length Relationships

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

N.RN.A.2

A.SSE.A.2

Multiply and divide radicals. Rationalize the denominator.

5

N.RN.A.2

A.SSE.A.2

Add and subtract radicals.

Topic B: Right Triangle Trigonometry

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.

Topic C: Applications of Right Triangle Trigonometry

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.

Topic D: The Unit Circle

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.

Topic E: Trigonometric Ratios in Non-Right Triangles

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

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.

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.

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).

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.

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.

Similarity, Right Triangles, and Trigonometry

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.

Similarity, Right Triangles, and Trigonometry

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.

Similarity, Right Triangles, and Trigonometry

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.

Similarity, Right Triangles, and Trigonometry

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.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.

Similarity, Right Triangles, and Trigonometry

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

Similarity, Right Triangles, and Trigonometry

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).

Similarity, Right Triangles, and Trigonometry

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).

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.

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.

Foundational Standards

Congruence

G.CO.C.10

Congruence

G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Expressions and Equations

8.EE.A.2

Expressions and Equations

8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.B.5

Expressions and Equations

8.EE.B.5 — Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6

Expressions and Equations

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Geometry

4.G.A.1

Geometry

4.G.A.1 — Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
6.G.A.1

Geometry

6.G.A.1 — Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.4

Geometry

8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.B.6

Geometry

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7

Geometry

8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8

Geometry

8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Ratios and Proportional Relationships

7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

Future Standards

Trigonometric Functions

F.TF.A.1

Trigonometric Functions

F.TF.A.1 — Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F.TF.A.2

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.
F.TF.A.3

Trigonometric Functions

F.TF.A.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, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
F.TF.A.4

Trigonometric Functions

F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
F.TF.B.5

Trigonometric Functions

F.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 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.
F.TF.B.6

Trigonometric Functions

F.TF.B.6 — Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
F.TF.B.7

Trigonometric Functions

F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
F.TF.C.8

Trigonometric Functions

F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
F.TF.C.9

Trigonometric Functions

F.TF.C.9 — Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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.