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.
- 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  and an angle whose measure is  ?
What do  and  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 and are the degree 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 vice-versa. Since the denominators of both ratios contain the hypotenuse, then sin =  which is the same as the 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  ].
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 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  ?
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 = 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 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 = sin(90 - )] to determine that:
- If sin
 = 0.32 , then cos (90 - )= 0.32 and if cos  = 0.68, then sin (90 - ) = 0.68.
- If sin A = 0.41 and cos B = 0.41 then and are complements.
The student further explains that and 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. |
