Getting Started 
Misconception/Error The student does not understand the relationship between the sine and cosine of the acute angles of a right triangle. 
Examples of Student Work at this Level The student writes that . The student:
 Finds and compares it to .
 Is unable to answer the question because the “triangle is not isosceles.”
 Incorrectly identifies the sine and/or cosine ratio.
 Explains that cannot equal because the trigonometric functions and/or angle measures are not the same.
 Explains that it cannot be determined whether or not sin = because angle measures and/or side measures are not given.

Questions Eliciting Thinking What are you asked to compare?
How is the sine of an angle defined? How is the cosine of an angle defined?
What do and represent?
How can you mathematically describe and ?
Do you think it is possible for sin to equal cos ? Why or why not? 
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 relationship between 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 [i.e., if and are the measures of the acute angles of a right triangle, then the side opposite the angle of measure is the same as the side adjacent to the angle of measure (and viceversa)]. Since the denominators of both ratios contain the hypotenuse, which is the same as (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 ].
Review with the student the difference between finding a ratio of sides and a degree measure of an angle. Provide the student with problems some of which involve finding an unknown length and some involving finding an unknown angle measure. Ask the student to explicitly determine what needs to be found and then solve the problem. Review with the student when a calculator is needed and not needed to find a value. Be sure the student understands that the trigonometric ratios are defined in reference to the acute angles of the right triangle.
Consider implementing other MFAS tasks for GSRT.3.7. 
Making Progress 
Misconception/Error The student is unable to explain why the sine and cosine of complementary angle measures are equal. 
Examples of Student Work at this Level The student writes that sin = cos and offers a very minimal explanation that lacks important detail such as:
 The ratios are the same.
 It is a right triangle.

Questions Eliciting Thinking Can you explain to me why the ratios are the same?
Will the ratios be the same in all right triangles or just in some?
Is there anything special about and ? If sin = cos , what must be true about and ? 
Instructional Implications 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 [i.e., if and are the measures of the acute angles of a right triangle, then the side opposite the angle of measure is the same as the side adjacent to the angle of measure (and viceversa)]. Since the denominators of both ratios contain the hypotenuse, sin is the same as cos (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 that ].
Provide problems in which the student must apply this understanding such as:
 If sin = cos , what is the value of ?
 If sin , what is the value of ?
 The sine of is equal to what trigonometric ratio of ?
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 writes that sin = cos because the side adjacent to one acute angle of a right triangle is the same side as the side opposite the other acute angle; therefore, the sine of of one acute angle will always be equal to the cosine of the other acute angle.
The student also writes that this relationship exists because angles and are complementary. The student understands that and .

Questions Eliciting Thinking If , what is the value of x?
If + = , what does equal?
What happens to the value of sin as increases from 0 to 90? What happens to the value of cos as increases from 0 to 90?
Are there any angle measures for which sin = cos , 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.
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. 