Right Triangles and Trigonometry

Lesson 2

Unit 4: Right Triangles and Trigonometry

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.

Objective

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

Common Core Standards

Core Standards

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.

Foundational Standards

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.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.

Criteria for Success

Define the Pythagorean theorem as, “If a triangle is a right triangle, then the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.”
Define the converse of the Pythagorean theorem as, “If the sum of the squares of two shorter sides is equal to the square of the third side, then the triangle is a right triangle.”
Prove the Pythagorean theorem using the similarity of the two triangles formed by subdividing the larger triangle using an altitude.
Use the Pythagorean theorem to find missing measures in right triangles.
Use altitudes and the Pythagorean theorem to identify missing measures (directly or algebraically) and in the development of proofs.

Tips for Teachers

This lesson formalizes students’ understanding of the Pythagorean theorem from Grade 8 Geometry.
Anchor Problem #1 will require the teacher to do a significant amount of modeling to prove the Pythagorean theorem and its converse. Reference EngageNY, Geometry, Module 2, Lesson 24: Teacher Version for more guidance on this.
Students may need to review simplifying radicals before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.

Anchor Problems

Problem 1

Given the subdivided right triangle below, show that $${a^2+b^2=c^2}$$.

Write an expression in terms of $$c$$ for $$x$$ and $$y$$.
Write a similarity statement for the three right triangles in the diagram.
Write a ratio that shows the relationship between side lengths of two of the triangles.
Prove the Pythagorean theorem.
Prove the converse of the Pythagorean theorem.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 24 — Image from Teacher Version: Opening

Geometry > Module 2 > Topic D > Lesson 24 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Find $${RS}$$.

Guiding Questions

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References

CK-12 8.3 Using Similar Right Triangles — Image from Example 3

8.3 Using Similar Right Triangles is made available by CK-12 under the CC BY-NC 3.0 license. © CK-12 Foundation 2017. Visit them at ck12.org. Accessed Feb. 28, 2018, 11:10 a.m..

Problem Set

The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

Include problems with descriptions and note ratios of right triangles with an altitude drawn.
Include more basic practice with Pythagorean theorem, such as Illustrative Mathematics, “8.G.B” specifically “Applying the Pythagorean Theorem in a Mathematical Context”.

Mathematics Vision Project: Secondary Mathematics Two Module 6: Similarity and Right Triangle Trigonometry — Lesson 7 "Pythagoras by Proportions" (p. 39-43)

Illustrative Mathematics Finding the Area of an Equilateral Triangle

Target Task

Problem 1

In each row in the table below are the lengths of the legs and hypotenuse of different right triangles. Find the missing side lengths in each row in simplest radical form.

$${\mathrm{Leg}_1}$$	$${\mathrm{Leg}_2}$$	Hypotenuse
15		25
15	36
3		7
100	200

References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 24 — Problem Set, Question #1

Problem 2

In right triangle $${ABC}$$ with $${\angle C}$$ a right angle, an altitude of length $$h$$ is dropped to side $${\overline{AB}}$$ into segments of length $$x$$ and $$y$$. Use the Pythagorean theorem to show $$h^2=xy$$.

References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 24 — Problem Set, Question #5

Right Triangles and Trigonometry

Lesson 2

Unit 4: Right Triangles and Trigonometry

Topic A: Right Triangle Properties and Side-Length Relationships

1

2

3

4

5

Topic B: Right Triangle Trigonometry

6

7

8

9

10

Topic C: Applications of Right Triangle Trigonometry

11

12

13

Topic D: The Unit Circle

14

15

Topic E: Trigonometric Ratios in Non-Right Triangles

16

17

18

19

Objective

Common Core Standards

Core Standards

Similarity, Right Triangles, and Trigonometry

Foundational Standards

Expressions and Equations

Geometry

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem Set

Target Task

Problem 1

References

Problem 2

References

lesson 1

lesson 3