An introduction to right triangle trigonometry
This lesson uses geometry to introduce students to trigonometric ratios and right triangle trigonometry. In this lesson, students will use their own knowledge of similar triangles to discover the trigonometric ratios sine, cosine and tangent in right triangles. Of course, the students will not recognize these ratios as being trigonometric ratios, but through class discussion will address these definitions. This lesson will also incorporate the ratios found in 45-45-90 and 30-60-90 triangles as well as introduce tan x as the slope of a line. Finally, there will be a short discussion of finding trig values for angles using a calculator.
Objectives:
-introduce trig ratios sine, cosine and tangent
-help students understand ratios in special triangles
-relate tan x as the slope of a line
-address finding values for trig functions using a calculator
NCTM Standards:
-Geometry through similar triangles and trig ratios
-Algebra through discussion of slopes of lines
-Reasoning and Proof through similar triangle activity
-Communication through group work and assignment write-up
-Connections between algebra and geometry
Teacher Activities / Student activitiesIf students are not already in groups, break them into groups of 3-4. Pose the following questions to the groups (using the above diagram)
-Which of the above triangles are similar, assuming right angles at C, E, G, and I? Write proofs for these similarities.
-From your similar triangles, the ratios of which sides are always equal?
-What angle is involved in the similarity of all of the triangles?
-Which of the ratios you found corresponds to th slope of segment AH? Justify your answer.
Discuss students’ results and introduce trig ratios, relating it to the angle question above. All of the ratios involve angle A. Talk about sin A, cos A and tan A in a right triangle.
Talk about the above slope question noting that tan A gives you the slope of AH.
Have students work on a worksheet in groups on finding the ratios for triangles with specific sides. On this worksheet, also have specific angles marked. For example, incorporate the ratios discussed for 45-45-90 and 30-60-90 triangles in Lesson 3.
Finally, come together as a whole class and discuss the worksheet, as well as finding values for specific angles with a calculator.
Assign homework and journal entry. / Students work in groups on the questions. They may have trouble with similar triangles, some may have forgotten what it means, which would require a brief explanation from the teacher or a peer of similar triangles and the angle-angle theorem.
Contribute to class discussion.
Students explain which ratio they found to be the slope of AH, some may not make the immediate connection to tan A, stress this point. It is important for students to know that tan x is a ratio.
Work on worksheet, may have trouble with specific angles. For example, might not be comfortable with writing tan 45 = 1.
Contribute to discussion and work on their calculators.
Possible Accommodations: Most of this lesson will be accessible to most students regardless of their individual needs because there is enough variety and activity to keep students engaged. Also, group work will help students sort out some of the mathematical problems they are having. Students that have trouble writing, would have the option to record their journal assignment on audio tape.
Assessment: For this portion of the lesson, students will be assessed on their class participation as well as their written work. Their worksheets will be collected and graded, partially for completion. Also, their journal will be graded for effort and demonstration of critical thinking.