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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#: 73142
Patterns in the 30-60-90 Table
Students are asked to use 30-60-90 triangle relationships to observe and explain the relationship between sin 30 and cos 60 (or sin 60 and cos 30).
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, sine, cosine, right triangle, complementary angles, special right triangle, 30-60-90
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_PatternsInThe306090Table_Worksheet.docx
MFAS_PatternsInThe306090Table_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 Patterns in the 30-60-90 Table worksheet.
2. The teacher asks follow-up questions, as needed.
Note: It is recommended that the student not use a calculator for this task because the student is asked to leave answers in simplest radical form.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to clearly and correctly describe the relationship between the sine of an angle and the cosine of its complement.
Examples of Student Work at this Level
The student:
Describes the relationship between sin
and cos
(or sin
and cos
) as “complementary.” page 1 of 4 Describes the relationship between sin
and cos
(or sin
and cos
) as “congruent.” Incorrectly completes the table so cannot discern the relationship.
Questions Eliciting Thinking
What does complementary mean? To what does this term usually apply? Can sin 30° and cos 60° be complements?
What does congruent mean? To what does this term usually apply? Can sin 30° and cos 60° be “congruent”?
How is the sine of an angle defined? How is the cosine of an angle defined?
What is sin 30°? Can you demonstrate how you determined the sine ratio for 30°?
What is cos 60°? Can you demonstrate how you determined the cosine ratio for 60°?
Look closely at the chart. Do you see any ratios that are the same?
Instructional Implications
Review the definitions of the sine and cosine ratios. Provide 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 of an angle and the cosine of its complement in terms of the definitions of the ratios. For example, if
right triangle, then the side opposite
is the same as the side adjacent to
and
are the acute angles of a
and vice-versa. Since the denominators of both ratios contain the hypotenuse, then sin A
is the same as the cos B (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.
Moving Forward
Misconception/Error
The student is unable to explain why the sine of an angle is equal to the cosine of its complement.
Examples of Student Work at this Level
The student correctly completes the table and observes that sin 30° = cos 60° and sin 60° = cos 30°, but is unable to write a complete explanation of these relationships.
The student provides a minimal response and is unable to elaborate. For example, the student says the ratios are equal because:
The angles are complementary.
They are the same.
page 2 of 4 Questions Eliciting Thinking
Can you explain why the sine of an angle is equal to the cosine of its complement? Why does this relationship hold for complementary angles?
Why do you think some of the ratios are the same? How can the definitions of sine and cosine help you determine why some ratios are the same?
Instructional Implications
Guide the student to explain the relationship between the sine of an angle and the cosine of its complement in terms of the definitions of the ratios. For example, if
and
are the acute angles of a right triangle, then the side opposite
is the same as the side adjacent to
and vice-versa. Since the denominators of both ratios
contain the hypotenuse, then sin A is the same as the cos B (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
, what is the value of
= cos
The sine of
. What is the value of
?
?
is equal to what trigonometric ratio of
?
Consider implementing other MFAS tasks for G-SRT.3.7.
Almost There
Misconception/Error
The student’s explanation is not precisely written.
Examples of Student Work at this Level
The student correctly completes the table, observes that sin 30° = cos 60° and sin 60° = cos 30°, and provides an essentially correct explanation. However, the
explanation:
Lacks precision.
Includes incorrect use of terminology.
Questions Eliciting Thinking
What did you mean by “the opposite”? Can you be more explicit?
What about the denominators of the ratios? Did you address the denominators in your explanation?
What did you mean by the ratios are “congruent”? Is “congruent” the correct word to use in this instance?
Instructional Implications
Provide feedback to the student concerning any errors or omissions. Remind the student that writing a clear and concise explanation requires using correct mathematical
terminology. For this question, the mathematical terminology is related to the definition of the sine and cosine functions. Prompt the student to explain using an illustration
of a specific example such as the given diagram or to draw a more general right triangle without identifying specific angle measures or side lengths. Encourage the student
to combine the definition of the trigonometric ratios with an example/diagram to write a thorough explanation for why this relationship occurs.
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 correctly completes the table, observes that sin 30° = cos 60° and sin 60° = cos 30°, and provides a correct explanation.TFor example, the student writes:
page 3 of 4 The definition of sin A =
and cos B =
angles, the side (leg) opposite of
. For any right triangle ABC, where
, is the same as the side (leg) adjacent to
is the right angle, and
,
are each acute
(and vice-versa). This produces equal numerators. Since the denominators of both
ratios contain the hypotenuse, they have the same value. Therefore, when
(
are complementary angles), sin A° = cos B° (and sin B° =
and
cos A°).
Questions Eliciting Thinking
Do you think this relationship holds for any right triangle? So what must be true of the measures of the angles in order for sin A = cos B?
If sin
°= cos , what is the value of
What happens to the value of sin
° as ? If sin
= cos
°. What is the value of ?
increases from 0 to 90 degrees? What happens to the value of cos
Are there any angle measures for which sin
= cos
and sin
= cos
as
increases from 0 to 90 degrees?
? If so, what are the measures of the angles and what is the ratio of the sides?
Instructional Implications
Allow the student to use diagrams or geometric software to illustrate why sin
increases and cos
decreases as
increases from 0 to 90 degrees.
Provide the student with the definitions of the secant, cosecant, and cotangent ratios. Ask the student to write each of the six trigonometric ratios for
triangle given on the worksheet. Have the student pair the trigonometric ratios that are equal for
and
and
using the
. Ask the student if he or she sees a similarity in the names of the
ratios that are equal. Help the student remember this relationship by pointing out that the “co” in cosine, cosecant, and cotangent refers to the sine, secant, and tangent
of its complement, respectively. Consider implementing the activity Trigonometry Square 1 (http://illuminations.nctm.org/uploadedFiles/Content/Lessons/Resources/912/TrigDrills-AS-Square.pdf) to reinforce these equivalent ratios.
Consider implementing other MFAS tasks for G-SRT.3.7.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Patterns in the 30-60-90 Table 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.
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