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:

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 and are the acute angles of a right triangle, then the side opposite is the same as the side adjacent to and viceversa. 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 = cos , what is the value of ?
 If sin = cos . What is the value of ?
 The sine of is equal to what trigonometric ratio of ?
Consider implementing other MFAS tasks for GSRT.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.

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 viceversa. 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 = cos , what is the value of ?
 If sin = cos . What is the value of ?
 The sine of is equal to what trigonometric ratio of ?
Consider implementing other MFAS tasks for GSRT.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 GSRT.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:
The definition of sin A = and cos B = . For any right triangle ABC, where is the right angle, and , are each acute angles, the side (leg) opposite of , is the same as the side (leg) adjacent to (and viceversa). This produces equal numerators. Since the denominators of both ratios contain the hypotenuse, they have the same value. Therefore, when ( and are complementary angles), sin A° = cos B° (and sin B° = 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 ? If sin = cos °. What is the value of ?
What happens to the value of sin ° as increases from 0 to 90 degrees? What happens to the value of cos as increases from 0 to 90 degrees?
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 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 and using the triangle given on the worksheet. Have the student pair the trigonometric ratios that are equal for and . 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/TrigDrillsASSquare.pdf) to reinforce these equivalent ratios.
Consider implementing other MFAS tasks for GSRT.3.7. 