Course: High School TrigonometryUnit Title: Right Triangles and Trigonometric Ratios
Instructional Goals:
To have students understand the angles and sides of a right triangle, the Pythagorean theorem, and trigonometric ratios.
Performance Objectives:
1. Students will know the names of special angles and sides of a right triangle as well as their properties.
2. Students will learn the names of the six trigonometric ratios. Also, they will know how these ratios are formed by using the lengths of the sides of a right triangle and be able to use them to solve mathematical problems.
3. Students will be able to utilize the Pythagorean theorem to find the missing length of one of the sides of a right triangle.
Rationale:
Students need to see how the angles and sides of a right triangle relate to each other in order to understand the trigonometric ratios; these ratios must be mastered because they are the basis of this course and are necessary for further study in mathematics. In addition, students must understand the Pythagorean theorem because it is a fundamental theorem that instructs students how to relate the sides of a right triangle so that they are able to solve mathematical problems.
Content
1. The three angles of a triangle add up to 180 degrees.
2. A right triangle always has one special angle that is equivalent to ninety degrees. The other two angles are acute angles, which must equal ninety degrees when summed together.
3. There are three sides of a right triangle; sides are defined to be rays or line segments. There is one special side called the hypotenuse, which is opposite the right angle of ninety degrees. The other two sides are referred to as the legs.
4. The Pythagorean theorem is used to find the length of a side of a triangle. The formula is c2 = a2 + b2 with c being the hypotenuse of the right triangle.
5. The six trigonometric ratios can be used to find the degree of a certain angle of a right triangle.
Learning Activities:
1. Talk with students about the importance about knowing about triangles and their properties.
While doing this, remind students of the tiny triangles and value tables on their desks that they will use during the class period.
2. Explain to students that a triangle is generally made up of three sides. Tell students a side is a ray or line segment that is connected together. Also, tell them it has three angles, which must sum to 180 degrees.
3. Tell students that we are now only going to learn about the right triangle, a special triangle of trigonometry. Explain how a right triangle has a special angle of ninety degrees, and two other acute angles that are each less than ninety degrees. When these two angles are summed together, they equal to ninety.
4. Explain that of the right triangle's three sides, there is a special name for the one side opposite the right angle. This side is called the hypotenuse. The other two sides are called legs.
Have students mark the sides and angles of their triangle.
5. Present the Pythagorean theorem and stress its importance. First, tell students that the Greek philosopher Pythagoras, who lived in sixth century B.C., was responsible for creating this theorem.
6. Next, define what the Pythagorean theorem states; that is, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse ( c2 = a2 + b2 ).
7. Draw a right triangle on the chalkboard and show students the following problem:
8. Clearly identify the sides and angles of the triangle involved in this problem. Ask students what side we are finding.
9. Use the Pythagorean theorem to find the length of the missing side and solve the problem with help of students. Make sure to call on students so they can describe each step of the procedure.
Ask students if they have any questions.
10. Draw another right angle on the chalkboard and solve the following problem with students the same way as the previous one was solved.
11. Draw another triangle on the board. This time give students a few minutes to work with each other to solve the following problem.
12. Draw three boxes on the board. Along with the students, find the length of the missing side of the triangle that is shaped by the formation of the three boxes.
13. Explain to students that trigonometric ratios or functions are used to find the degree of an angle.
14. Write the ratios on the board and explain "A Happy Old Heap of Apples." Show how the secant, cosecant, and cotangent are reciprocals of cosine, sine, and tangent.
15. Draw a triangle on the board and solve the following problem with the help of students. Allow students to describe each step they are taking.
Explain how to find the value of the angle given by the trigonometric ratio from the table handed out to the students.
16. Present the following problem and solve as a class.
17. Present the following problems and give students a few minutes to work on them with a partner.
18. Conclude lesson by reviewing the important concepts taught during this class period. Next, pass out worksheet for students to work on and explain that they will complete the rest for homework.
Assessment:
1. Active student participation in solving the problems given during class.
2. Ability to describe how to use the Pythagorean theorem and the trigonometric ratios in solving for the sides and angles of a right triangle.
3. Ability to answer the questions on the worksheet.
Materials:
1. Calculators
2. Big triangle and little triangle
3. Sample Problems
4. Table of values of trigonometric ratios
5. Worksheet