Trigonometric Ratios

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Objective

SWBAT explain the relationship between the sine and cosine of complementary angles.

Big Idea

Students explore the relationship between trigonometric ratios in right triangles.

Do Now

7 minutes

In the Do Now, students answer questions based on concepts learned in the previous lesson. I use this Do Now to assess how much information students have retained and how much they understand. If I see that students still have misconceptions, I spend some time reteaching the concept after we go over the Do Now.

Mini-Lesson

10 minutes

In the Mini-Lesson, students are shown two triangles and asked when to use the Pythagorean theorem or trigonometric ratios in order to find the value of a variable (MP7). In the first triangle, the lengths of two sides are given and students have to find the length of the third side and in the second triangle, the length of one side and the measure of one angle are given. Based on prior knowledge, students can identify that they need to use the Pythagorean theorem to find the value of the variable in the first triangle. I have the students write down a statement explaining when to use the Pythagorean theorem.

Students can then identify that the trigonometric ratios are needed to find the value of the variable in the second triangle. They write down an explanation of how they know when to use trigonometric ratios. I ask the student if there would be another case for using trigonometric ratios. Students figure out that they can be given the lengths of two sides of a right triangle and use the trigonometric ratios to find the value of an acute interior angle.

We then work together to find the value of the variable in both triangles before students work independently on the activity.

Activity

21 minutes

In the Activity, students work independently to answer questions involving trigonometric ratios. The worksheet is designed to enable students to come to a conclusion about the relationship between the sine and cosine of acute angles in a right triangle (G.SRT.7).

In the first question, the students have to decide which method to use in order to find the value of a variable. They explain what method they used for one of the triangles in the second question. As students work, I circulate around the room. I look to see if the students used the correct trigonometric ratio and if they understand why to use it.

After about 5 minutes, I check in with the students. We go over the answers to questions 1 and 2 to ensure they are correct. If I see that students many students are having difficulty solving the problems, I show the students a method they can use to help figure out how to solve the problem. I show students three triangles with the letters SOH, CAH, and TOA written inside them. By covering up the letter that represents the missing information students can see whether they need to multiply, divide, or use the inverse trigonometry function. Then they continue working on the sheet.

After about 5 minutes, I check in with the students. We go over the answers to questions 1 and 2 to ensure they are correct. Then the students continue working on the problems.

If students complete all of the questions before the end of the activity, I have them go back to question 1 and find the measures of the other angles and sides in the triangle.

At the end of the activity, we go over questions 3 and 4. I address question 5 in the summary.

Summary

7 minutes

In the summary, we go over question 5 from the activity. I ask students to give their responses. Students identify that the sine of <A is equal to the cosine of <B and the cosine of <A is equal to the sine of <B. Then I have the students generalize this concept. I ask them what the relationship between <A and <B is. If students have difficulty verbalizing their response, I instruct them to refer back to the last question in the Do Now. The acute angles in a triangle are complementary and the sine of one acute angle is equal to the cosine of the other acute angle (G.SRT.7). If time permits, I have the students verify this by checking the sines and cosines of other sets of complementary angles.