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.
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.
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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 |
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.
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.
G.SRT.B.4
Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.
G.SRT.B.4
G.SRT.B.5
Define the relationship between side lengths of special right triangles.
N.RN.A.2
A.SSE.A.2
Multiply and divide radicals. Rationalize the denominator.
N.RN.A.2
A.SSE.A.2
Add and subtract radicals.
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.
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.
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°.
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°.
G.SRT.C.8
Solve for missing sides of a right triangle given the length of one side and measure of one angle.
G.SRT.C.8
Find the angle measure given two sides using inverse trigonometric functions.
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.
G.SRT.C.8
Solve a modeling problem using trigonometry.
F.TF.A.2
Define angles in standard position and use them to build the first quadrant of the unit circle.
F.TF.A.2
Use the first quadrant of the unit circle to define sine, cosine, and tangent values outside the first quadrant.
G.SRT.D.9
Derive the area formula for any triangle in terms of sine.
G.SRT.D.10
Verify algebraically and find missing measures using the Law of Sines.
G.SRT.D.10
Verify algebraically and find missing measures using the Law of Cosines.
G.SRT.D.11
Use side and angle relationships in right and non-right triangles to solve application problems.
Key: Major Cluster Supporting Cluster Additional Cluster
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