Download Right Triangle Relationships

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