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# Trigonometric Ratios

Lesson 2 of 19

## Objective: Students will be able to define trigonometric ratios of acute angles.

*51 minutes*

#### Warm Up and Homework Review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’s Warm Up- Trigonometric Ratios Warm Up asks students to solve a traditional Pythagorean Theorem word problem using a ladder against a wall. The Pythagorean Theorem will be important to today’s lesson so this is a great mini review.

I also use this time to correct and record the previous day's Homework.

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#### Trigonometric Ratios

*25 min*

I introduce this lesson by starting with a right angle, making sure the students remember the terminology legs and hypotenuse. I then place the right triangle in a circle centered on a coordinate plane. The intersection of the circle and the triangle is called (x,y). I then ask the students to talk in pairs and determine the length of the legs of the triangle. They should figure out that the horizontal side is the same length as the x value of the intersection and the vertical side is the same as the y value. We will call the hypotenuse r for radius. This will make a difference when we talk about radians.

My students saw the trig ratios in Geometry the previous year. Here we review their basic geometric definition using the position of angle *ө*. We also add the algebraic definition using x, y and r. Again, this will be important as they head into radians. Please watch my Video Narrative on prior knowledge.

Using a coordinate to represent (x,y)on a circle, we then find the sine, cosine and tangent ratios. They need to get r to find sine or cosine but I don’t tell them in advance (**Math Practice 1**). I walk around and give hints or individual aid to students that are really struggling. Pythagorean Theorem was the warm up so it shouldn’t be entire unfamiliar to them.

Next, they are asked to find sine, cosine or tangent given one of the ratios. This is similar to the previous problem except that they have to decode a trig ratio onto the graph, find the third side and then find a new trig ratio (**Math Practice 2**). This is a more complex task than the previous one. Again, I give the students an opportunity to work on this problem with no prompting. Scaffolding can be provided individually as necessary or to the group if a large number of them are floundering. Another example is provided to solidify the concept.

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#### Special Right Triangles

*13 min*

The remainder of the lesson focuses on the trig ratios found on the two special triangles 30-60-90 and 45-45-90 (**Math Practice 7**). One of the Pre-Calculus teachers at my school told me that one of the best things that students could bring to his class would be a larger understanding of trig, particularly as it relates to 45-45-90 and 30-60-90 triangles. I am choosing not to have my students memorize the basic trig functions but we will continue to visit these special triangles and by the end the students should have a solid understanding to those triangles. It would be good to talk about why these two are particularly important. The more connections students can make, the better they will understand them.

Please see the PowerPoint for detailed presentation notes.

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The Homework is a practice of the skills they learned in class. The best way for students to memorize the trigonometric ratios is to use them. This assignment will help them do that. It also asks the student to reproduce and explain how to find the trigonometric ratios off of the two special triangles.

#### Resources

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#### Exit Ticket

*3 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks the students to find sine, cosine and tangent given the (x,y) coordinate form on a graph.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Angle and Degree Measure
- LESSON 2: Trigonometric Ratios
- LESSON 3: Trigonometric Ratios of General Angles
- LESSON 4: Radians Day 1 of 2
- LESSON 5: Radians Day 2 of 2
- LESSON 6: The Unit Circle Day 1 of 2
- LESSON 7: The Unit Circle Day 2 of 2
- LESSON 8: Graphs of Sine and Cosine
- LESSON 9: Period and Amplitude
- LESSON 10: Period Puzzle
- LESSON 11: Transformations of Sine and Cosine Graphs
- LESSON 12: Graph of Tangent
- LESSON 13: Model Trigonometry with a Ferris Wheel Day 1 of 2
- LESSON 14: Model Trigonometry with a Ferris Wheel Day 2 of 2
- LESSON 15: Modeling Average Temperature with Trigonometry
- LESSON 16: Pythagorean Identity
- LESSON 17: Trigonometric Functions Review Day 1
- LESSON 18: Trigonometric Functions Review Day 2
- LESSON 19: Trigonometric Functions Test