Getting Started 
Misconception/Error The student does not understand the properties of a right triangle or the definitions of the trigonometric ratios. 
Examples of Student Work at this Level The student:
 Attempts to find angle measure of , but inputs into the calculator and writes that = 0.01.
 Incorrectly identifies the sine ratio as and writes that or as and writes sin = .
 Finds the sine ratio of and writes sin = .
 Labels the lengths of the sides of the triangle and then finds sin .
 Finds the measure of , and writes that .
 Writes sin is the reciprocal of the cos , that is, .
 Labels the lengths of the sides of the triangle incorrectly and, consequently, describes the sin incorrectly.

Questions Eliciting Thinking What are you trying to find when asked, “What is sin ”?
Are you looking for the ratio of sides or the angle measure?
How would the question have been written if you were to find the angle measure?
What ratios of sides of a right triangle are represented by sine, cosine, and tangent?
Based on the given cosine ratio, which side has a length of 3 and which side has a length of 5?
Can you demonstrate how you determined was the sine ratio for ?
Will you demonstrate for me how you arrived at your answer?
Do you think sin could equal cos ? Why or why not? 
Instructional Implications Review with the student the vocabulary associated with right triangles (e.g. right, acute, and complementary angles, opposite and adjacent sides, legs, and hypotenuse). Provide the student with a right triangle with side lengths given. Have the student identify the sine, cosine, and tangent ratios for each acute angle. If needed, include right triangles in different orientations. Remind the student that the trigonometric ratios are only defined for acute angles in right triangles. Given one of the trigonometric ratios (e.g., ) have the student draw a right triangle and label the length of each side and the measures of the angles ( and ). Have the student write the other two trigonometric ratios (e.g., sine and tangent) of and the three trigonometric ratios of .
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 of which involve 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.
Using a calculator, have the student complete a chart of the sine and cosine ratios of several pairs of complementary angles. Guide the student to observe that the sine of an acute angle is equal to the cosine of its complement. Use the context of a right triangle to demonstrate why the ratios are equal. Emphasize that 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 cosine of one acute angle is equal to the sine of the other acute angle. Help the student remember this relationship by pointing out that the “co” in cosine refers to the sine of its complement. 
Moving Forward 
Misconception/Error The student does not understand the relationship between and . 
Examples of Student Work at this Level The student:
 Does not respond to the second question.
 Writes that both angles are congruent.
 Writes that both angles are equal to and are therefore equal to each other.
 Writes that both angles are .
 Writes that the angles are supplementary.

Questions Eliciting Thinking If both acute angles in a right triangle are equal, what is true about their measures?
What do you know about the side lengths of a 454590 triangle?
If the side opposite one angle in a triangle is greater than the side opposite another angle, what must be true about the first angle?
If two angles are congruent in the same triangle, what type of triangle is it? Is this triangle isosceles?
What is the definition of supplementary angles? 
Instructional Implications Have the student calculate the measure of and using his or her calculator. If needed, review with the student how to find an angle measure given its cosine ratio. Have the student verify that the measures of the two acute angles are different. Provide the student with several other triangles with given side lengths. Using a calculator, ask the student to make a chart listing the measures of the two acute angles in each triangle. Guide the student to observe that within each triangle, the sine of one acute angle is equal to the cosine of its complement. Help the student remember this relationship by pointing out that the “co” in cosine refers to the sine of its complement. If needed, review with the student the definition of complementary angles.
If the student writes 0.6 as his or her answer to the first question, make sure he or she understands that a ratio, , is also an appropriate representation for that answer. 
Almost There 
Misconception/Error The student does not include that and are complementary in his or her reasoning about what is true regarding the two angle measures. 
Examples of Student Work at this Level The student writes:
 The angle measures must be different.
 The angles must lie in the same right triangle.
 That must be greater than because it lies opposite the longer side.

Questions Eliciting Thinking Is there anything else that you know about the measures of the two acute angles of a right triangle? 
Instructional Implications Guide the student to observe that the two acute angles of a right triangle are complementary. If needed, review with the student the definition of complementary angles.
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.
If the student writes 0.6 as his or her answer to the first question, make sure he or she understands that a ratio, , is also an appropriate representation for that answer. 
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 = . The student understands that 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 ratio of sides will be the same.
The student also writes that angles and are complementary. The student understands that and that .

Questions Eliciting Thinking The sin 15° = cos x°. What is the value of x?
If , what does equal?
If , are there any angle measures for which sin = cos ? If yes, what are they? If not, why not? 
Instructional Implications 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.
Challenge the student to use his or her understanding of the Pythagorean Theorem to explain why . 