ALEX Lesson Plan


"I Saw the Sine"

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  This lesson provided by:  
Author:Morgan Boyd
  General Lesson Information  
Lesson Plan ID: 35580


"I Saw the Sine"


This lesson will provide information that will prove the concept of sine and cosine is equal to the complementary angles of a right triangle. The lesson will examine the proper techniques for writing trigonometric ratios. The lesson will enhance background knowledge of proportions as well as use the terminology of means and extremes.

This lesson results from the ALEX Resource Gap Project.

 Associated Standards and Objectives 
Content Standard(s):
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
35. Discover and apply relationships in similar right triangles.

a. Derive and apply the constant ratios of the sides in special right triangles (45o-45o-90o and 30o-60o-90o).

b. Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.

c. Explain and use the relationship between the sine and cosine of complementary angles.

d. Demonstrate the converse of the Pythagorean Theorem.

e. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
Unpacked Content
Evidence Of Student Attainment:
  • Given a collection of right triangles, discover and apply relationships in similar right triangles.
  • Derive and apply the ratios of the sides of the original triangles to the ratios of the sides of the similar triangles.
  • Communicate observations made about changes (or no change) to such ratios as the length of the side opposite an angle to the hypotenuse, or the side opposite the angle to the side adjacent, as the size of the angle changes or in the case of similar triangles, remains the same.
    Summarize these observations by defining the six trigonometric ratios.
  • Explain why the two smallest angles must be complements.
  • Compare the side ratios of opposite/hypotenuse and adjacent/hypotenuse for each of these angles and discuss conclusions.

Given a contextual situation involving right triangles,
  • Create a drawing to model the situation.
  • Find the missing sides and/or angles using trigonometric ratios.
  • Find the missing sides using the Pythagorean Theorem.
  • Use the above information to interpret results in the context of the situation, including finding the areas of regular polygons.
Teacher Vocabulary:
  • Side ratios
  • Trigonometric ratios
  • Sine
  • Cosine
  • Tangent
  • Secant
  • Cosecant
  • Cotangent
  • Complementary angles converse
Students know:
  • Techniques to construct similar triangles.
  • Properties of similar triangles.
  • Methods for finding sine and cosine ratios in a right triangle (e.g., use of triangle properties: similarity. Pythagorean Theorem. isosceles and equilateral characteristics for 45-45-90 and 30-60-90 triangles and technology for others).
  • Methods of using the trigonometric ratios to solve for sides or angles in a right triangle.
  • The Pythagorean Theorem and its use in solving for unknown parts of a right triangle.
Students are able to:
  • Accurately find the side ratios of triangles.
  • Explain and justify relationships between the side ratios of a right triangle and the angles of a right triangle.
Students understand that:
  • The ratios of the sides of right triangles are dependent on the size of the angles of the triangle.
  • The sine of an angle is equal to the cosine of the complement of the angle.
  • Switching between using a given angle or its complement and between sine or cosine ratios may be used when solving contextual problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.35.1: Define trigonometric (sine, cosine and tangent) ratios for acute angles, complementary angles, and Pythagorean Theorem.
GEO.35.2: Simplify, multiply, and divide radicals.
GEO.35.3:Discuss the relationship between sine and cosine angles within a triangle.
GEO.35.4: Solve equations using trigonometric ratios.
GEO.35.5: Apply properties of similarity to demonstrate the trigonometric ratios of right triangles.
GEO.35.6: Use Pythagorean Theorem to find the missing side of a right triangle.
GEO.35.7: Create an equation using the given information of a right triangle.
GEO.35.8: Identify the parts of a right triangle.
Examples: legs, hypotenuse, right angle.

Prior Knowledge Skills:
  • Demonstrate how to find square roots.
  • Identify right triangles.
  • Solve problems using the Pythagorean Theorem.
  • Recognize ordered pairs (x, y).
  • Recall how to name points on a Cartesian plane using ordered pairs.
  • Identify right triangles.
  • Solve problems using the Pythagorean Theorem.
  • Discuss strategies for solving real-world and mathematical problems.
  • Recognize examples of right triangles.
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Apply properties to find missing angle measures.
  • Identify a transversal.
  • Identify exterior, interior, vertical, adjacent, alternate interior, alternate exterior, and corresponding angles.
  • Identify attributes of triangles.
  • Define exterior, interior, vertical, adjacent, alternate interior, alternate exterior, corresponding, and transversal angles.
  • Identify right and straight angles.
  • Identify all types of angles.
  • Identify proportional relationships.
  • Locate/use scale on a map.
  • Define scale, scale drawings, length, area, and geometric figures.
  • Recall how to find unit rates using ratios.
  • Define unit rate, proportions, area, length, and ratio.
  • Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
  • Define area, special quadrilaterals, right triangles, and polygons.
  • Recognize and demonstrate that two right triangles make a rectangle.
  • Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
  • Explain how to find the area for rectangles.
  • Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
  • Apply area formulas to solve real-world mathematical problems.
  • Recognize polygons.
  • Restate real-world problems or mathematical problems.
  • Calculate unit rate or rate by using ratios or proportions.
  • Create a ratio or proportion from a given word problem, diagram, table, or equation.
  • Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
  • Form a ratio.
  • Solve a proportion using part over whole equals percent over 100.
  • Identify a proportion from given information.
  • Calculate a proportion for missing information.
  • Create a proportion or ratio from a given word problem.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.

Local/National Standards:


Primary Learning Objective(s):

The student will be able to determine the sides of a right triangle.

The student will demonstrate knowledge of ratios of a right triangle for sine and cosine.

The student will compare ratios for sine and cosine for right triangles.

Additional Learning Objective(s):

The student will calculate sine and cosine values using a calculator.

The student will calculate angles given the values of sine and cosine using a calculator.



 Preparation Information 

Total Duration:

31 to 60 Minutes

Materials and Resources:

Resources for Teacher Activation and Preview

1. Basic Trigonometry video from Khan Academy

2.  Finding-Trigonometric-Ratios.pdf is PDF file worksheet in the attachments/resources section. This resource was downloaded from Kuta Software. The teacher will need to make copies for all students unless the teacher is using Google classroom.

3. Writing ratios for sine, cosine, and tangent from Khan Academy

4. PowerPoint in the attachments/resources section for the bell ringer and the exit slip

5. Worksheet for the accelerated students in the attachments/resources section. Copies need be made for students that finish early.

Students will need one sheet of paper and a pencil.

Technology Resources Needed:

Interactive whiteboard with computer/projector for the teacher and the students

iPad, Chromebooks, or MacBooks for each student for intervention (revisiting websites) and submitting classwork on Google classroom

Wi-Fi and Google classroom for all students and teacher

Calculators with trig functions for all students


Teacher Background

The teacher must have the knowledge to show students how to use a calculator to find sine, cosine, and tangent values. All calculators are different when it comes to inputting values for angles. Casio calculators ask for the angle measure first then trig ratio. Texas Instrument calculators will need the trig ratio first then the angle measure. The teacher and students will need to experiment with the calculators to understand how each type works. The teacher will need to preview the website and activities with Khan Academy. The website and program are a great resource for teachers. The teacher will need to be ready to explain and show “means and extremes” to the students.

Student Background

The students will need to understand ratios, unit rates, and proportions. The students will need knowledge of complementary angles as well as properties of a right triangle. The student should be able to use the Pythagorean Theorem to calculate missing sides of a right triangle.  



  1. As the students enter the classroom, the Bell Ringer should be posted on the interactive whiteboard (The bellringer can be found on the first slide of the PowerPoint located in the attachments/resources section).
  2. Select students at random to answer the bell ringer. (Answers: Hypotenuse is 5, one leg is 3 and the other leg is 4.)
  3. Introduce the video from Khan Academy (“Basic Trigonometry”)
  4. The teacher can stop the video for questioning and comprehension.


  1. After the video, the teacher should introduce the saying “sock-a-tow-a” (SOH-CAH-TOA).
  2. Show the students how to use the calculator to find sine, cosine, and tangent. Press the trig ratio key and then the angle.
  3. Use the calculator to find the angle from the decimal form of the trig ratio. Press INV key then trig ratio key.
  4. Explain that the letters in front stand for a particular trig ratio of the right triangle.
    1. S – sine
    2. C – cosine
    3. T – tangent
    4. O – opposite side of the given angle
    5. A – adjacent side of the given angle
    6. H – hypotenuse
    7. Use the Bell Ringer and given the bottom angle to write the trig ratios in fraction form and decimal form.
      1. 3/5 and 0.6 are the Sine using the bottom angle
      2. 4/5 and 0.8 are the Cosine using the bottom angle
      3. Show the video “Writing the ratios for sine, cosine, and tangent”.
      4. Using the angles of a right triangle, show the students that the two angles will be acute. The sum of the two acute angles must be 90 degrees, which makes them complementary.
      5. Using the definition of proportions, the sine of one acute angle will equal the cosine of the other acute angle
      6. The students will make a table and compare ratios.

From the bell ringer triangle…Draw the triangle on the smart board

Sine x


Sine y


Cosine x


Cosine y


Students compare ratios: sine x = cosine y

Students compare ratios: cosine x = sine y

The students have proven that the trig ratios (sine/cosine) of opposite complementary angles of a right triangle are equal.


  1. Assign the problems from the PDF worksheet in the attachments/resources section.
  2. The teacher can upload the worksheet in Google classroom, then the students can access it and submit the answers.
  3. If the teacher does not have access to technology, then he/she will need to make copies for each student.
  4. The students will complete the worksheet for homework.

Exit Slip

As the students begin the assignment, place the second slide from the PowerPoint in the attachment/resources section on the smart board. Ask the students to complete the problem on their own paper and turn it in as they leave the classroom.

**Some files will display in a new window. Others will prompt you to download.

Assessment Strategies


The teacher will assess the students as the video is playing as well as soliciting answers from students.

The teacher will make an informal assessment by grading the worksheet that is turned in or submitted through Google classroom.


The teacher can make a formal assessment by using the exit slip at the end of class. The students will be able to complete the assignment with 100% mastery. The exit slip is the second slide from the PowerPoint in the attachment/resources section.


Copy the worksheets named "Accelerated" in the attachments/resources section. These worksheets are for the students that finish early. 


The students that need accommodating and extra reinforcement will receive one-on-one tutorial. Also, the teacher can pair the students with a peer tutor from the class.

View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.