Download Sine, Cosine, and Tangent: The Leap from Special

Primary Type: Lesson Plan
Status: Published
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Resource ID#: 48711
Sine, Cosine, and Tangent: The Leap from Special
Triangles to the Unit Circle
Students will discover the connection between finding trigonometric ratios (sine, cosine, and tangent) using special right triangles and the unit circle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Instructional Time: 55 Minute(s)
Freely Available: Yes
Keywords: sine, cosine, tangent, trigonometric ratios, trigonometry, special triangles, unit circle, radian
Instructional Design Framework(s): Direct Instruction, Demonstration, Structured Inquiry (Level 2), Guided
Inquiry (Level 3), Writing to Learn
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
unit circle.pdf
Find the indicated value 1.docx
Find the indicated value 2 Answer Key.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
The student will use special triangles to connect trigonometric ratios to the unit circle.
Student will find missing x and y coordinates on the unit circle by finding trigonometric values geometrically and by using symmetry about x and y axis.
Students will know the sign value of trigonometric functions on the unit circle no matter what quadrant angle lies in.
Prior Knowledge: What prior knowledge should students have for this lesson?
Radian measure
Definition of sine/cosine/tangent ratios, (cos
=
, and cos
=
)
Properties of similar triangles
Pythagorean Theorem
Guiding Questions: What are the guiding questions for this lesson?
How do we find trigonometric values geometrically? (soh,cah,toa)
How do the side lengths of similar triangles affect the trigonometric values? (they don't)
How do the side lengths relate to the x and y coordinates? (because the length of the radius is one on a unit circle, the coordinates correspond to the side lengths of
a special triangle with hypotenuse of one)
page 1 of 3 What are the trigonometric values when the angle is
(By symmetry, if one knows sin(x) or cos(x), it is straightforward to
obtain their trigonometric values.)
Teaching Phase: How will the teacher present the concept or skill to students?
1. (15-20 min) Teacher begins with activity sheet reinforcing prior knowledge on Pythagorean Theorem, similar triangles, and finding sine, cosine, and tangent values
geometrically. Students should see that the value of sine, cosine, and tangent for the angle 30 is the same no matter what the length of sides are.
2. Teacher should now ask how this activity is relevant to the unit circle. Discuss what a unit circle is, why it is called "unit" (radius is 1). Suggest students to move to
one of the triangles with a radius of one, so that the
angle is placed at origin.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
1. (15 min) Next, teacher should ask students to continue process with the other special angles, write down what they notice for each value, how it relates to the
length of x and y on the coordinate plane, and explain how it connects to the geometric way of finding values, to using the unit circle.
2. (10 min) Finally ask students to fill in the rest of the unit circle (emphasize importance of being accurate for they will refer to it often). Remind students of the
properties of the coordinate plane and be mindful of sign values.
3. Papers should be turned in for review by the teacher and returned next day so students have an accurate chart for reference.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Students should finish completing the unit circle after guided practice through quadrant one.
Here are two free KUTA software worksheets that you could use:
Exact Trig Values of Special Angles
Trigonometric Ratios
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students were required to write their conclusions on the 2nd activity. Teacher should read these and give individualized feedback through written comments on sticky
notes, or the paper itself and turn back in next day so students can be assisted on what they have learned.
Summative Assessment
Given a blank unit circle, students will fill in missing angles and coordinates, and use that information to find the sine, cosine, and tangent values for specified angles.
Student has reached learning goal if there are few mechanical mistakes (forgetting a negative/positive, or 2*3=5) and no error in the concept (how to find the values
for these functions)
Formative Assessment
Teacher will walk around and assess students' understanding:
How to find trigonometric functions geometrically (soh-cah-toa)
How to find the missing side of a special triangle
How to simplify complex fractions (fractions within a fraction)
Ability to work with both radian and degree measures of arc length
This is the worksheet and the answer key
Feedback to Students
Students receive one-on-one feedback during activities and guided practice.
Students may talk with and provide feedback to one another. Through this process they get a better understanding on how special triangles and the unit circle
connect, how the x value corresponds to cosine, y to sine, and become more comfortable using radian measure.
The circle activity sheet should be turned in for the teacher to review to ensure accuracy before students use it for a reference sheet. This allows the teacher to
correct on the spot, reinforce prior content, and get an idea on where the students are in the lesson.
How to connect the special triangle with radius one, to the unit circle, and its sides to the length of x and y on the coordinate plane,
Values of trigonometric functions when
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations: Teacher may have a few circles partially filled for students with special needs. Visual aids (pre-cut paper triangle that teacher can place in unit
circle, as described in lesson, on board/overhead). If a student needs more help, have a student that is not struggling help him/her through peer teaching.
Extensions: Students who finish early and seem to grasp the concept can be challenged by having them graph sin
as the domain, and sin
on an x and y coordinate plane with
as the range.
Further Recommendations: soh-cah-toa:
The teacher may need to guide students with first activity (finding trig values geometrically).
But, more importantly guide them through the second activity as they transfer the geometric concept to the unit circle concept, and relate that idea into a coherent
summary (because the radius is one with a unit circle, the x and y values could also represent leg values of a right triangle with hypotenuse of one).
Additional Information/Instructions
By Author/Submitter
Teachers may want to have plenty of blank unit circles for students who need extras.
The following standards of math practice may align with this lesson:
MAFS.K12.MP.3.1 - Construct viable arguments and critique the reasoning of others.
page 2 of 3 MAFS.K12.MP.6.1 - Attend to precision.
MAFS.K12.MP.8.1 - Look for and express regularity in repeated reasoning.
SOURCE AND ACCESS INFORMATION
Contributed by: caroline Campbell
Name of Author/Source: caroline Campbell
District/Organization of Contributor(s): Jackson
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.F-TF.1.3:
Description
Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the
unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x,
where x is any real number.
page 3 of 3