Math 2 Unit 5: Right Triangles and Trigonometry



Approximate Time Frame: 3 – 4 Weeks

Connections to Previous Learning:

In prior units, students studied the definition of similarity, and worked with proportions of corresponding sides and congruencies of corresponding angles. This unit will build on these ideas to specifically include the side ratios of right triangles and the definitions of trigonometric ratios for acute angles. Further, the Pythagorean Theorem (Grade 8) and its proof via similarity (previous unit) will be utilized in solving right triangles.

Focus of this Unit:

Students will use their knowledge of similarity to write the ratio of sides of right triangles as functions of its acute angles, defining the trigonometric ratios. They will use these definitions and the Pythagorean Theorem to solve right triangles.

Connections to Subsequent Learning:

Students will connect their knowledge of right triangle trigonometric ratios to the unit circle and graphic representations of the trigonometric functions.

|Desired Outcomes |

|Standard(s): |

|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 angle 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. |

|WIDA Standard: (English Language Learners) |

|English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. |

|English language learners benefit from: |

|explicit vocabulary instruction with regard to the study of angles in trigonometry. |

|tactile and virtual tools to study the relationships between sides and angles in right triangles. |

|Understandings: Students will understand that … |

|The ratios of the sides of right triangles are functions of the acute angles of the triangle. |

|The sine of an acute angle in a right triangle is equal to the cosine of that angle's complement (and vice versa). |

|The Pythagorean Theorem applies only to right triangles. |

|Essential Questions: |

|How does similarity give rise to the trigonometric ratios? |

|How do the trigonometric ratios of complementary angles relate to one another? |

|How can the Pythagorean Theorem be used to solve problems involving triangles? |

|Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) |

|*1. Make sense of problems and persevere in solving them. Students will solve problems in context that involve right triangles. |

|*2. Reason abstractly and quantitatively. Students reason about the ratios used to represent relationships between sides and angles. |

|3. Construct viable arguments and critique the reasoning of others. |

|4. Model with mathematics. |

|*5. Use appropriate tools strategically. Students will use calculators to evaluate trigonometric values and to solve for sides of a right triangle. |

|*6. Attend to precision. While solving problems involving right triangles, students will attend to the precision of their answers. They will use appropriate language to describe their measurements and calculations.|

|*7. Look for and make use of structure. Students will set appropriate ratios of the sides of right triangles equal to the sine, cosine, or tangent of an angle. Also, students will recognize when it is appropriate |

|to use the Pythagorean Theorem. |

|*8. Look for express regularity in repeated reasoning. Students will recognize that trigonometric ratios arise from the ratio of sides of similar right triangles. |

|Prerequisite Skills/Concepts: |Advanced Skills/Concepts: |

|Students should already be able to: |Some students may be ready to: |

|Use the Pythagorean Theorem to solve for a missing side of a right triangle with integer-valued |See the complementary angle trigonometric identities as horizontal transformations. |

|sides. |Define the reciprocal trigonometric functions. |

| |Define the inverse trigonometric functions. |

| |Solve for missing sides in non-right triangles, using the Law of Sines and Law of Cosines. |

| |Recognize and use the slope of a line as the tangent of its angle of elevation. |

|Knowledge: Students will know… |Skills: Students will be able to … |

|The trigonometric function definitions of sine, cosine, and tangent as ratios of the sides of a |Use the trigonometric ratios and knowledge of special right triangles to determine the sine, cosine, and tangent values|

|right triangle. |of 30º, 45º, and 60º without the assistance of technology. |

| |Apply the Pythagorean Theorem to problems involving right triangles. |

| |Solve for the angles in a right triangle, given at least two sides. |

| |Solve for the missing sides of a right triangle, given either two sides or one acute angle and one side. |

|Academic Vocabulary: |

| | | | |

|Critical Terms: | |Supplemental Terms: | |

|Hypotenuse | |Pythagorean triple | |

|Opposite side | |"solve" a triangle | |

|Adjacent side | |Complementary angles | |

|Sine | |Special right triangles (30-60-90, 45-45-90) | |

|Cosine | | | |

|Tangent | | | |

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