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Sine and Cosine Relationship between
Complementary Angles
Resource ID#: 130872
Primary Type: Lesson Plan
This document was generated on CPALMS - www.cpalms.org
This is a lesson on the relationship between the Sine and Cosine values of Complementary
Angles.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera, Computer for Presenter, Interactive Whiteboard,
LCD Projector, Overhead Projector, Scientific Calculator
Instructional Time: 1 Hour(s) 30 Minute(s)
Keywords: Sine, Cosine, Complementary Angles
Instructional Component Type(s): Lesson Plan
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
Sine and Cosine Relationship Guided Practice.docx
Sine and Cosine Relationship Guided Practice answer key.docx
Sine and Cosine Relationship Independent Practice.docx
Sine and Cosine Relationship Independent Practice 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?
Students will calculate the sine and cosine of both acute angles in a right triangle and explain why the sine
of one acute angle is equal to the cosine of the complementary angle.

Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be able to:
o
o
o
o

use the Pythagorean Theorem and its converse
define sine and cosine as trigonometric ratios
identify and calculate complementary angles
use a scientific calculator to determine the sine and cosine of angles
Guiding Questions: What are the guiding questions for this lesson?
What is the relationship between the sine and cosine values of complementary angles in a right
triangle? Why does this relationship occur?

Teaching Phase: How will the teacher present the concept or skill to students?
Due to prior knowledge for this lesson, the teaching phase will be 10 minutes. The teacher will review the
following definitions and concepts with the students: complementary angles, sine and cosine ratios, and the
Pythagorean Theorem. The teacher can ask student volunteers to explain to the class verbally and then have
another student draw a diagram at the board that represents each verbal explanation.
Distribute Sine and Cosine Relationship - Guided Practice.
The teacher will then find/verify Sine A, Cosine A, Sine D and Cosine D in problems 1 and 2 from the
guided practice sheet.

Guided Practice: What activities or exercises will the students complete with teacher
guidance?
Students will continue on the guided practice sheet, finishing the first two right triangles, then completing
the final three triangles completely. The guided practice phase will be 25 minutes.
This practice can be done individually or in groups, based on the teacher's discretion. If the students do well
in the review, let them work individually. Students who have difficulty with determining the Sine and
Cosine ratios or using the Pythagorean Theorem should be grouped with students who are not having
difficulty.

Independent Practice: What activities or exercises will students complete to reinforce
the concepts and skills developed in the lesson?
Distribute Sine and Cosine Relationship - Independent Practice.
The independent practice worksheet is to be completed in class (if time allows). The independent practice
should be completed in 45 minutes.
All five right triangles should be addressed. Students should complete the table for each triangle with the
Sine and Cosine of the acute angles, identify which values are equal, and explain why they are equal. The
final problem has connecting right triangles, and the student will use the trigonometric formulas to find the
sine ratios for both acute angles in each triangle. The student will then apply the pattern discovered in this
lesson (the sine and cosine ratios of complementary angles in a right triangle are congruent) to determine
the cosine ratio of the complementary angle.

Closure: How will the teacher assist students in organizing the knowledge gained in the
lesson?
The teacher will collect the assignments 10 minutes before the end of the period. The teacher will then ask
students to solve one of the first five problems. Once the students have properly found the Sine and Cosine
Ratio for both acute angles in a right triangle, the teacher will then ask students for their explanation on
why Sine and Cosine of Complementary Angles are equal.
The teacher will grade the Independent Practice and return it to the students with notations and corrections
at the beginning of the next class.

Summative Assessment
Students will complete Sine and Cosine Relationship - Independent Practice. First to be completed will be a
table with sides, side lengths, and sine and cosine values for the acute angles of five different triangles.
Then, students will identify which values are the same and explain why the sine and cosine values of
complementary angles are the same. Finally, students will find the sine values of six acute angles in a series
of connecting right triangles, then determine the cosine values of those acute angles without using the ratios
to work them out.

Formative Assessment
After a general review to start the class the teacher will walk around the room to assess the students' prior
knowledge on:
1.
2.
3.
Sine and Cosine Ratios
Complementary Angles
Pythagorean Theorem
While walking the room, the teacher will ask the students questions to assess how well they understand
these skills. The teacher can then group students that fully understand the skills with those that need some
help.

Feedback to Students
o
o
o
Students will receive group feedback during the guided practice, the teacher will answer questions
on all skills.
Students will receive one-on-one feedback during the guided and independent practice, the teacher
will let individual students know if they are correct or need to rework certain problems.
Students will be allowed to talk to their fellow classmates to get peer to peer feedback during the
guided and independent practice.
ACCOMMODATIONS & RECOMMENDATIONS

Accommodations:
The problems on the worksheets can be modified for students having difficulty recognizing the pattern. For
example, only using Pythagorean triples, only angles with integer measures, and labeling the hypotenuse of
each triangle will reduce distractions and allow all students to learn the concept and practice the skill.

Extensions:
o
o


Use triangles that do not use Pythagorean triples as the length of the sides.
Use triangle side lengths that will create fractions with radicals in both the numerator and
denominator.
o Have students try to determine the relationship between the Tangent value of Complementary
Angles.
o Label the sides of a 30-60-90 triangle.
Suggested Technology: Document Camera, Computer for Presenter, Interactive Whiteboard, LCD
Projector, Overhead Projector, Scientific Calculator
Special Materials Needed:
o
o
o

An Interactive Whiteboard or a document camera will aide the teacher in presenting this lesson
with out having to draw all the examples by hand
Students should have access to scientific calculators
Copies of "Sine and Cosine Relationship - Guided Practice" (one per student) and "Sine and
Cosine Relationship - Independent Practice" (one per student)
Further Recommendations:
This lesson could be done in a group setting, but it is just as effective with the students working
individually. Prepare cards ahead of time for each of the Trigonometric Ratios in the Guided Practice (Sine
A, Cosine A, Sine B, Cosine B, etc) as well as cards with a variety of answers (including answers that are
incorrect). Place the answers around the room. Give the students the Sine and Cosine cards and have the
students go stand at the value for their trigonometric ratio. Once each group is paired up, have them explain
how they got their value and why it is equal to their partner's. If students get the wrong answer, have them
explain to you how they got their answer and then help them see the error in their work.
The cards for the first problem in the guided practice could included correct answers 3/5 and 4/5 and
possible errors 3/4, 4/3, 5/3, and 5/4.
Additional Information/Instructions
By Author/Submitter
This lesson aligns with the following mathematical practices:
MAFS.K12.MP.2.1 - Reason abstractly and quantitatively.
MAFS.K12.MP.6.1 - Attend to precision.
MAFS.K11.MP.8.1 - Look for and express regularity in repeated reasoning.
SOURCE AND ACCESS INFORMATION
Contributed by: Kevin Whelan
Name of Author/Source: Kevin Whelan
District/Organization of Contributor(s): Miami-Dade
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.3.7:
Description
Explain and use the relationship between the sine and cosine of
complementary angles.