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
H
F
B
D
A
I
C
E
G
Teacher Activities
Student activities
If 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
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.
-
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 Contribute to class discussion.
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.
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.
Have students work on a worksheet in
Work on worksheet, may have trouble with
groups on finding the ratios for triangles
specific angles. For example, might not be
with specific sides. On this worksheet, also comfortable with writing tan 45 = 1.
have specific angles marked. For example,
incorporate the ratios discussed for 45-4590 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.
Contribute to discussion and work on their
calculators.
Assign homework and journal entry.
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.