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Primary Type: Formative Assessment Status: Published This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas! Resource ID#: 73185 Right Triangle Relationships Students are given the sine and cosine of angle measures and asked to identify the sine and cosine of their complements. Subject(s): Mathematics Grade Level(s): 9, 10, 11, 12 Intended Audience: Educators Freely Available: Yes Keywords: MFAS, sine, cosine, right triangles, complementary angles Resource Collection: MFAS Formative Assessments ATTACHMENTS MFAS_RightTriangleRelationships_Worksheet.docx MFAS_RightTriangleRelationships_Worksheet.pdf FORMATIVE ASSESSMENT TASK Instructions for Implementing the Task This task can be implemented individually, with small groups, or with the whole class. 1. The teacher asks the student to complete the problems on the Right Triangle Relationships worksheet. 2. The teacher asks follow-up questions, as needed. TASK RUBRIC Getting Started Misconception/Error The student does not understand the concept of trigonometric ratios. Examples of Student Work at this Level The student: Subtracts the given values from 90. page 1 of 3 Writes a response that does not make mathematical sense. Attempts to find the indicated values in the first and second questions using a calculator but is unsuccessful. Questions Eliciting Thinking What is the relationship between an angle whose measure is What do and and an angle whose measure is ? represent? What do 0.32 and 0.68 represent? How is the sine of an angle defined? How is the cosine of an angle defined? Can you draw a right triangle to illustrate these definitions? Instructional Implications Review the definitions of the sine and cosine ratios. Provide problems in which the student is asked to calculate an unknown length and an unknown angle measure of a right triangle. Emphasize the definitions of the sine and cosine ratios and the distinction between an angle measure and the length of a side. Be sure the student understands the meaning of both the input and output of a calculator when attempting to calculate the sine of an angle or the inverse sine of a ratio of sides. Provide experiences that will help the student develop an understanding of the relationship between the sine and cosine of complementary angle measures. For example, show the student a variety of right triangles in various orientations and ask the student to identify the sine and cosine ratios of each acute angle. Have the student organize the results in a way that makes it possible to observe the relationships among the ratios. Guide the student to explain the relationship between the sine and cosine ratios of the acute angles in terms of the definitions of the ratios. For example, if the angle of measure sin = and are the degree measures of the acute angles of a right triangle, then the side opposite is the same as the side adjacent to the angle of measure which is the same as the cos and vice-versa. Since the denominators of both ratios contain the hypotenuse, then (and vice versa). Help the student remember this relationship by pointing out that the “co” in cosine refers to the sine of its complement. Guide the student to generalize this relationship to all complementary angle pairs [i.e., and ]. Provide problems in which students must apply this understanding such as: If sin If sin = cos = cos The sine of , what is the value of ? . What is the value of ? is equal to what trigonometric ratio of ? Consider implementing other MFAS tasks for G-SRT.3.7. Making Progress Misconception/Error The student does not understand the relationship between the sine and cosine of complementary angle measures. Examples of Student Work at this Level The student correctly finds the indicated values in the first two problems. However, the student is unable to correctly explain the relationship between two angles for which the sine of one is equal to the cosine of the other. For example, the student Does not recognize the angles as complements. Says the angles must be equal in order to be complementary. Provides a nonmathematical explanation such as the sine and cosine “can’t get away from each other.” Questions Eliciting Thinking What is the relationship between the measures of and ? page 2 of 3 What is the definition of complementary angles? Must complementary angles be equal? Can you explain what you meant by this? What does this mean mathematically? Instructional Implications Review the relationship between the sine and cosine of complementary angle measures. If needed, guide the student to use this relationship to answer the questions in this task rather than using a calculator. Provide additional problems in which students can use this relationship to answer questions such as: If sin If sin = cos = cos The sine of , what is the value of ? . What is the value of ? is equal to what trigonometric ratio of ? Consider implementing other MFAS tasks for G-SRT.3.7. Got It Misconception/Error The student provides complete and correct responses to all components of the task. Examples of Student Work at this Level The student uses the relationship between the sine and cosine of complementary angles [i.e., cos If sin = 0.32 , then cos (90 - )= 0.32 and if cos If sin A = 0.41 and cos B = 0.41 then The student further explains that and and = 0.68, then sin (90 - = sin(90 - )] to determine that: ) = 0.68. are complements. must be complements since the sine of an angle is equal to the cosine of its complement (and vice versa). Questions Eliciting Thinking What is the relationship between the sine of an angle and the cosine of its complement? How did you use this relationship to answer these questions? If , what does equal? Are there any angle measures for which sin = cos and sin = cos ? If so, what are the measures of the angles and what is the ratio of the sides? Instructional Implications Challenge the student to use right traingle relationships to explain why and the . Consider implementing other MFAS tasks for G-SRT.3.7. ACCOMMODATIONS & RECOMMENDATIONS Special Materials Needed: Right Triangle Relationships worksheet SOURCE AND ACCESS INFORMATION Contributed by: MFAS FCRSTEM Name of Author/Source: MFAS FCRSTEM District/Organization of Contributor(s): Okaloosa Is this Resource freely Available? Yes Access Privileges: Public License: CPALMS License - no distribution - non commercial Related Standards Name MAFS.912.G-SRT.3.7: Description Explain and use the relationship between the sine and cosine of complementary angles. page 3 of 3