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Classroom Interactions

Title of Lesson: Inverse Trigonometric Functions with GatorsUFTeach Students’ Names: Zack Brenneman, Julie WalthallTeaching Date and Time: November 14, 2013 1:00 – 2:45Length of Lesson: 50 minutesCourse / Grade / Topic: Pre-Calculus Honors / 11th-12th / Inverse Trigonometry Source of the Lesson: Larson & Hostetler Pre-Calculus with Limits textbook. Resource ID#: 48711 CPalms (Caroline Campbell’s “Sine, Cosine, and Tangent” lesson plan). Inmath.com (for Cosby picture). https://mathtestpreparation.com/Lessons/TrignInverseSineFun.aspx for sine graph on ppt.

Embedding Strategies Based on Observations:Based on the readings and what happened in class, I am including the following teaching strategies with these

students because… in order to comprehend inverse trigonometric functions, students must understand basic trigonometric properties, ratios, concepts, etc. These strategies will help re-enforce previously learned trigonometric concepts and allow students to extend their knowledge to learn about inverses. These strategies will aid students in in-depth discovery of the new material because working in groups will extend each individuals knowledge; learning through new media will keep students engaged; and teaching through questioning will allow students to explore the concepts for themselves rather than being fed the information.

Recommended strategy Reason for selecting this strategy

Describe where in Lesson Plan this strategy would

best fitGroup collaboration Exploring brand new

concepts can be difficult and intimidating to students individually. So, allowing students to work in groups will build confidence and promote deeper thinking and effort.

During the exploration. At this time, students will work together to discover new terminology, new notation, and new concepts. By working together during this part of the lesson, students will be more engaged and confident before the explanation.

Modeling and “Non-textbook” learning material

Students seem bored with traditional learning materials such as their textbook and note taking. Using the smart board to draw or having students use graphing calculators (when using difficult angle measurements or non-traditional trig ratios.) will allow them to explore concepts involving trigonometry, degrees, radians, and angle measure

During the engagement, explanation, and elaboration. The explanation will use an online image projected on the smartboard. The explanation will use the smartboard for the PowerPoint, notes, and examples. The elaboration will use the smartboard where a new problem is projected for students to read in front of them and

https://mathtestpreparation.com/Lessons/TrignInverseSineFun.aspx


in a new fun way. work out a solution. Questioning in Teaching Asking questions to the

students and letting them provide the answers requires students to take an active approach to learning. This is crucial when learning new topics for the first time, like in this lesson.

During the exploration, the teacher will ask probing questions to each student. During the explanation, the teacher will call on students for their answers from the exploration worksheet. While doing so, the teacher will ask them open-ended questions to ensure the students learn the correct reasoning behind the solutions.

Common Core State Standards (CCSS) / Next Generation Sunshine State Standards (NGSSS):

Standards Number Benchmark Description Cognitive Complexity

MACC.912.F-TF.2.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Level 2: Basic Application of Skills & Concepts

MACC.912.F-TF.2.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Level 2: Basic Application of Skills & Concepts

Concept Development:From Pre-Calculus with Limits textbook, in order for a function to have an inverse, it must be one-to-one (pass the horizontal line test). By restricting the domain of trigonometric functions, we can define the inverse functions. Using the following table, it is possible to find inverses when the domain is restricted as below. Definitions of the Inverse Trigonometric FunctionsFunction: Domain: Range: y = arcsin x iff sin y = x -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2y = arccos x iff cos y = x -1 ≤ x ≤ 1 0 ≤ y ≤ πy = arctan x iff tan y = x -∞ < x ≤ ∞ -π/2 < y ≤ π/2

Inverse Properties of Trigonometric FunctionsIf -1 ≤ x ≤ 1 and -1 ≤ x ≤ 1, then sin(arcsin x) = x and arcsin(sin y) = yIf -1 ≤ x ≤ 1and 0 ≤ y ≤ π, then cos(arcos x) = x and arc(sin y) = yIf x is a real number and -π/2 < y < π/2, then tan(arctan x) = x and arctan(tan y) = y.


In general, when evaluating the inverse, it helps to remember the phrase, “the arcsin/arccos/arctan of x is the angle whose sine/cosine/tangent is x.”

Performance Objectives Students will be able to restrict the domain of a trigonometric function. (Evaluated in explanation /

evaluation) Students will be able to solve trigonometric equations. (Evaluated in exploration / explanation /

evaluation) Students will be able to solve inverse trigonometric functions. (Evaluated in exploration / explanation /

evaluation) Students will be able to apply trigonometry to real world problems. (Evaluated in elaboration)

Materials List 27 exploration 1 worksheets 27 TI inspire calculators (for elaboration) 27 evaluation worksheets 1 PowerPoint with slides corresponding to 5E’s

Advance Preparations Teachers will have PowerPoint uploaded and projected onto SmartBoard before class begins Teachers will have made copies of each worksheet before class Teachers will have materials organized and ready to be passed out Teachers will have nametags prepared and ready for students to place on desks

Safety I anticipate potential problems with the calculators. (Off-task behavior such as creating random graphs or

playing games if the TI-Inspire provides it.) Students will be instructed to keep the calculators on their desks and use them for intended use only. No other significant concerns


ENGAGEMENT Time: 3 MinutesWhat the Teacher Will Do What the Teacher Will Say

(include Probing Questions)Student Responses and Potential Misconceptions

Switch to Slide 1 of the Power Point. Teacher will introduce himself/herself and welcome the class.

Welcome students! My name is Mr. Brenneman / Miss Walthall.

Good Afternoon!

Teacher will give overview of lesson topic

Today, we will be studying Trigonometric Functions more in depth, specifically, relating to their inverses!

Oh great, more concepts with trig functions :/

Sounds interesting!Switch to Slide 2 of the Power Point. Teacher will introduce the engagement activity (Bill Cosby Image.)

To begin, I want to show you all an example of some math humor.

Why is there a person up on the board?!

Teacher Response: Because this person helps with the logic of this picture.

Teachers will ask if any students know who this man is.

Who knows who this man on the right is?

I don’t know

[Bill Cosby]Teachers will ask a question related to this image.

Take a minute and see if you can reason why this equation is true.

Hmmmm…Something to do with tangent and sine. Where is the cosine though?

Teacher will ask students to explain what their reasoning.

Who wants to share with the class what they found?

[I will! Basically, sinb/tanb = cosb or “cosby” which is the name of the person in the picture.]

No Idea.

Teacher Response: Well we should know that sinb/cosb = tanb. So how would I manipulate this equation so we have sinb/tanb?

Teacher will segue into exploration While we are in a mindset of trig functions, we will start today’s main lesson.

Sounds good! Wonder what’s in store for today’s lesson!

EXPLORATION 1 Time: 10 MinutesWhat the Teacher Will Do What the Teacher Will Say

(include Probing Questions)Student Responses and

Misconceptions

Switch to Slide 3 of the PowerPoint. Teachers will open with a starting question.

What can you tell me about sin (π/2)?

[Sin(π/2) = 1]

Sin(π/2) = 0.Teachers will ask the reverse of the starting question.

What if I wanted to go backwards? That is, what angle gives us 1 as its

[π/2 or 90 degrees]π/2 or 90 degrees.


sine value?I don’t know.

Teacher Response: Well, angles would be 0, π/2, π, etc. radians (or its corresponding degrees). So which of these angles would give us 1 as it’s sign value? (Point to the sine value 1 if necessary.)

Teachers will denote how to correctly write the sine inverse function.

So we just found sine inverse of 1. We can denote this as sin-1(1) orarcsin(1).

So does this imply:sin-1(1) = (1/sin)(1)

Teacher Response: No it does not. Although it looks like we are raising sine to the negative first power, this is just how we represent the inverse sine function. Please do not confuse this as 1 over sin.

Teachers will ask another question related to the topic.

Now let me ask you, what can you tell me about sin(π/4)?

[Sin(π/4) = /2]

Sin(π/4) = 1Teachers will ask the reverse of the same question.

What if I wanted to go backwards? That is, what angle gives us /2 as its sine value? (Or what is sin-1( /2)?)

[π/4 and 3π/4]

π/4

Teacher Response: Are there any other angles which gives us a sine value of /2?

Teachers will elaborate on the previous question.

What could I do to have a unique inverse? That is, have only one value to represent sin-1( /2),

[You can restrict the domain so that we are only looking at sine values from – π/2 to π/2.]

Not sure.

Teacher Response: What could I do to this Unit Circle so that we see sine’s full range of values? (Without repeating a value of sine.)

Teachers will go in to detail about restricting the domain so that the function has a unique inverse.

Notice if I were to restrict the domain to the interval [-π/2, π/2], sinx would take on its full range of values, from -1 ≤ sinx ≤1. So in this interval, we say sinx has a unique inverse function called the inverse sine function. (Denoted arcsinx or sin-1(x).)

What about from π/2 to 3π/2?

Teacher Response: All the sine values in this region have already been accounted for, so we will not be using this interval when finding the inverse sine function.

Teachers will explain instructions For this next part, you and your So we are doing similar concepts,


for the Exploration Worksheet. group mates will be given additional practice with inverse sine functions, as well as exploring inverse cosine and inverse tangent functions. Once you have been given a worksheet, you may begin working. If you have a question, raise your hand, and we will assist you.

but now with the inverse cosine and inverse tangent function?

Teacher Response: Correct

Teachers will ask students if they have any questions.

What questions do you have? How long do we have to work on this?

Teacher Response: About 5 Minutes

Teachers will ask students to make groups of 4 or 5 with the classmates sitting next to them. Once groups are made, teachers will pass out the worksheets.

Here is your worksheet. You will have about 5 minute to complete this.

Misconceptions:

Students work with their friends on the other side of class.

EXPLANATION 1 Time: 7 MinutesWhat the Teacher Will Do Teacher Directions and

Probing/Eliciting QuestionsStudent Responses and

Misconceptions

*After 5 Minutes have passed* Teachers will gain students attention and begin to go over the exploration worksheet.

May I have everyone’s attention? We will now be going over this worksheet.

Misconceptions:

Students continue to work on assignment.

Students continue talking to their classmates.

Switch to Slide 4 of the Power Point. Teachers will ask what students answered for number 1 and 2.

Who wants to share what they got for question 1 and 2?

[π/6 and π/4 respectively.]

π/3 and π/2

Teachers will ask students how they came to this answer.

How did you get π/6 and π/4 respectively?

[The interval where sine has an inverse is from –π/2 to π/2. Sin-1 (1/2) is π/6 and sin-1 ( /2) = π/4, since -π/2 ≤ π/6, π/4 ≤ π/2.]

Teachers will ask what students answered for number 3 and 4.


[π/2 and 5π/6 respectively.]

π/3 and π/2.Teachers will ask students how they came to this answer.

How did you get π/2 and 5π/6 respectively?

[I took a guess and made the interval of the inverse cosine function from [0,π]. So cos-1(0) = π/2 and cos-1(- |3/2) = 5π/6, since0 ≤ π/2, 5π/6 ≤ π.]


No Idea!

Teacher Response: Don’t worry, we will go over definitions soon and will come back to this question.

Teachers will ask what students answered for number 5 and 6.


[0π and –π/4 respectively.]

π/2 and 7π/4.Teachers will ask students how they came to this answer.

How did you get 0π and –π/4 respectively?

[I took a guess and made the interval of the inverse tangent function from [-π/2, π/2]. So tan-1(0) = 0 and tan-1(-1) = -π/4, since –π/2 ≤ -π/4, 0 ≤ π/2.]

No Idea!

Teacher Response: Don’t worry, we will go over definitions soon and will come back to this question.

Switch to Slide 5 of the Power Point. Teachers will go over the definition of the inverse sine function.

We somewhat already learned about the restrictions we need to put on the inverse sine graph so that we have a unique inverse. Notice that the range of the inverse sine function is:[-π/2, π/2] and the domain is [-1,1].

Note: Teachers will go back to the first two questions of the exploration worksheet and go over the questions again.

What do you mean when you say the range is:[-π/2, π/2] and the domain is [-1,1]?

Teacher Response: In order for sine to have an inverse function, we must restrict the range from[-π/2, π/2]. In this interval, sinx takes on its full range of values (-1 ≤ x ≤ 1).

Teachers will go over the definition of the inverse cosine function.

For the cosine inverse function, we restrict the range to [0, π]. Also notice that the domain of the inverse cosine function is from [-1,1].

Note: Teachers will go back to the next two questions of the exploration worksheet and go over the questions again.

What do you mean when you say the range is [0, π] and the domain is [-1,1]?

Teacher Response: In order for cosine to have an inverse function, we must restrict the range from[0, π]. In this interval, cosx takes on its full range of values (-1 ≤ x ≤ 1).

Teachers will go over the definition of the inverse tangent function.

For the tangent inverse function, we restrict the range to [-π/2, π/2]. Also notice that the domain of the

What do you mean when you say the range [-π/2, π/2] and the domain is [-∞, ∞]?


inverse tangent function is from (-∞,∞).

Note: Teachers will go back to the next two questions of the exploration worksheet and go over the questions again.

Teacher Response: In order for tangent to have an inverse function, we must restrict the range from[-π/2, π/2]. In this interval, tanx takes on its full range of values (-∞ ≤ x ≤ ∞).

Why is it from [-∞, ∞] instead from [-1,1].

Teacher Response: Well we are looking at the domain in this function. Notice that there is a horizontal asymptote on the line y = -π/2 and y = π/2. So as x keeps increasing or decreasing, it will keep approaching that line, but never touch it.

Depending on time and student understanding, teachers will Switch to Slide 6 of the PowerPoint and break down the correlations between the sine, cosine and tangent graphs with their corresponding inverse graphs.

Notice for the sine graph, if I were to restrict the domain from [-π/2, π/2], it would be one to one (or pass the Horizontal Line Test.)

Reflecting this over the line y = x (which gives us the sine inverse function), we see that the domain is restricted to [-1,1] and the range is restricted to [-π/2, π/2].

Wouldn’t the range go from (-∞, ∞) reflecting it over the line y = x?

Teacher Response: Yes it would! But in order to be a function, it must pass the Vertical Line Test. If we let the graph go to negative and positive infinity, it would not pass the Vertical Line Test.



Misconceptions

Teachers will present a beginning problem based on inverse properties of trigonometric functions.

If I were to ask you to evaluate the expression sin [sin-1(1/2)], what would your guess be?

[1/2]

2

I don’t know. We haven’t had a problem like this before.

Teacher Response: Well we know from before that sin-1 is the inverse of sin. For real numbers, something times its inverse is usually what? How can we apply that concept to sin[sin-1(x)].


Switch to Slide 7 of the Power Point (Exploration 2). Teachers will present 3 more questions about properties of trigonometric functions.

You may use the back of your worksheet to answer these three questions. Work in your groups. You will have about 4 minutes to work on this and afterwards, we will go over these problems.

I don’t have any room on the back of my paper. I used up all the space. What should I do?

Teacher Response: We have some spare pieces of paper you can use.

Teachers will gain students attention and begin to explain the three problems.

May I have everyone’s attention? We will be going over these three problems now.

Misconceptions:

Students continue to work on the worksheet.

Students continue talking with their group mates.

EXPLANATION 2 Time: 5 MinutesWhat the Teacher Will Do Teacher Directions and

Probing/Eliciting QuestionsStudent Responses and

Misconceptions

Teachers will ask students what their answers were for the first question.

Who wants to share how they evaluated tan [arc tan (-5)]?

[It is just -5 since -5 lies in the domain of the arc tan function,; hence, the inverse property applies.]

5 because it must always be positive.

No idea.

Teacher Response: Because -5 lies in the domain of the arc tan function, the inverse property applies, and you have tan (arc tan (-5)) = -5

Teachers will ask students what their answers were for the second question.

Who wants to share how they

evaluated arc sin [sin ( )]?

[-π/3 since 5π/3 is coterminal with –π/3.]

There is no solution because it must be in the form sin (arc sin (x)).

5π/3

I don’t know

Teacher Response: In this case, 5π/3 does not lie within the range of the arcsine function, (which is – π/2 ≤ y ≤ π/2). However, 5π/3 is coterminal with 5π/3 - 2 π = - π/3


which does lie in the range of the arcsine function, and you have:arc sin (sin (5π/3)) = arc sin (sin (-π/3)) = -π/3

Teachers will ask students what their answers were for the third question.

Who wants to share how they evaluated cos [cos-1 ( )]?

[It is not defined since π is not defined.]

π

I don’t know.

Teacher Response: The expression cos(cos-1 (π)) is not defined because cos-1 (π) is not defined. Remember that the domain of the inverse cosine function is [-1,1].

Switch to Slide 8 of the Power Point. Teachers will provide definitions for inverse properties of trigonometric functions.

If -1 ≤ x ≤ 1 and – π/2 ≤ y ≤ π/2, then sin(arc sin x) = x and arc sin(sin y) = y.

NOTE: Refer to the Power Point for the remaining two definitions.

Misconceptions:

Students think the domain and range can be any real number for the inverse trig functions to apply.

Students use sin (x) instead of sin (y) or arc sin (y) instead of arc sin (x). (Same situation for cosine and tangent.)

Teachers will ask students what questions they have based on Exploration #2.

What questions do you have regarding these concepts?

Will these inverse properties apply for any arbitrary values of x and y?

[Teacher Response: They do not. For instance, arc sin (sin (3π/2)) = arc sin (-1) = -π/2 ≠3π/2. In other words, the property arc sin (sin y) = y is not valid for values of y outside the interval [-π/2,π/2].]



Misconceptions

Switch to Slide 9 of the Power Point. Teachers will go over a question which students should already know how to do.

Here, we are given this is a right triangle. We also know one of the angles is 32.9 degrees as well as one of the side lengths equaling 17. How would I solve for the missing side x?

[We let tan (32.9) = x/17. Solving for x, we have x = 17 tan(32.9) ~ 11. Hence x = 11.]

No idea.

Teacher Response: We should


know that tan (32.9) = x/17. Solving for x, we have x = 17 tan (32.9) ~ 11. Hence x = 11

Teachers will ask students how to take the same triangle and solve for the angle instead of the side length.

For this triangle, I am giving you both side lengths. We know that the angle is going to be the same since the side lengths equal the side lengths of the previous triangle. Your team’s goal is to try to find a way to get the angle measure. Take a few minutes and see if you can figure it out.

How are we supposed to solve this if we have never seen this problem before?

Teacher Response: Use what we did from the previous example (where the angle was given), go through the steps in the same fashion, and see what you come up with.

Time is up. Who wants to share what their group came up with?

[I will! We let tan θ = 11/17. Then we took tan-1 on both sides to get θ = tan-1 (11/17). Plugging this in to the calculator, θ = 32.91 degrees.]

No idea.

Teacher Response: If we knew the angle measure, we know to set the problem up like tan θ = 11/17. But I want to solve for θ. So what do I have to do to get θ by itself?

EXPLANATION 3 Time: 5 MinutesWhat the Teacher Will Do What the Teacher Will Say


Misconceptions

Teachers will ask students (or groups) if they found a way to the answer 32.9 degrees.

Time is up. Who wants to share what their group came up with?

[I will! We let tan θ = 11/17. Then we took tan-1 on both sides to get θ = tan-1 (11/17). Plugging this in to the calculator, θ = 32.91 degrees.]

No idea.

Teacher Response: If we knew the angle measure, we know to set the problem up like tan θ = 11/17. But I want to solve for θ. So what do I have to do to get θ by itself?

Teachers will ask students if they have any questions regarding this type of problem.

What questions do you have regarding this problem?

What if everything was the same except 17 was the value of the hypotenuse. How would we answer that type of question?

[Teacher Response: In relevance


with the angle θ, we are given the opposite side length and we are given the hypotenuse side length. So we use the sine function. We set up this equation as: sin θ = 11/17 which implies θ = sin-1 (11/17) which implies θ = 70.4 degrees.]

ELABORATION Time: 5 MinutesWhat the Teacher Will Do Probing/Eliciting Questions Student Responses and

MisconceptionsSwitch to Slide 10 of the Power Point. Teachers will read the problem.

Now, let’s try and use what we have learned to solve a real world problem.

“A ladder is placed against a 40 foot high electric pole such that it touches the top of the pole. If the bottom of the ladder is 10 feet away from the base of the pole, what angle does the bottom of the ladder make with the ground?”

Take the next 3 minutes and let’s see what you get.

Can we work in groups?

Teacher Response: Try and answer this question individually.

After 3 minutes have passed, teachers will ask students for their answers.

3 Minutes are up. Who wants to share their answer?

[It is 75.96 degrees]

Teachers will go over the problem by visually representing it on the Smart Board.

I am first going to draw a pole which is 40 feet tall. From the problem, we know that a ladder is on a tilt touching the top of the pole. We also know that base of the ladder is 10 feet away from the pole. We set up this problem by saying tan θ = 40/10. This implies θ = tan-1(4) which implies θ = 75.96 degrees.

Misconceptions:

Why are you using tangent instead of cosine?

Teacher Response: Because if we look at this right triangle and we locate the angle θ, we are given the opposite side and the adjacent side. We use the tangent function with this set up.

EVALUATION Time: 5 MinutesWhat the Teacher Will Do Assessment Student Responses

Teachers will give instructions for the Evaluation.

Each of you will be given an evaluation. You are to work independently. You will have

Misconceptions:

Students work in groups.


roughly 5 minutes to complete this. Once you are finished, raise your hand and we will collect your paper.

Teachers will pass out evaluations to students.

Once you receive your paper, you may begin.

Misconceptions:

Students collaborate with one another.

Name:_______________________________

Exploration Worksheet

Evaluate the expression without using a calculator.


1) sin-1( /2) 2) arcsin (1/2)

3) cos-1( ) 4) arcos(0)

5) tan-1(-1) 6) arctan (0)

Name: Answer Key

Exploration Worksheet



1) sin-1( /2) π/4 2) arcsin (1/2) π/6

3) cos-1( ) 5π/6 4) arcos(0) π/2

5) tan-1(-1) -π/4 6) arctan (0) 0π

Name: ________________________________

Evaluation



Use the properties of inverse trigonometric functions to evaluate the expression.

3) sin [arcsin (½)]

What angle does θ represent? Assume this is a right triangle.

4)

Name: Answer Key

Evaluation


π/3


π/6

Use the properties of inverse trigonometric functions to evaluate the expression.

3) sin [arcsin (½)] ½

What angle does θ represent? Assume this is a right triangle. Round your answer to the nearest tenth. (Remember to have your calculator in Degree Mode).

4)

θ = 48.6°

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