Microsoft Word - S1_Syllabus.doc



Title of Lesson: Inverse Trigonometric Functions with Gators

UFTeach Students’ Names: Zack Brenneman, Julie Walthall

Teaching Date and Time: November 14, 2013 1:00 – 2:45

Length of Lesson: 50 minutes

Course / 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). (for Cosby picture). 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 fit |

|Group collaboration |Exploring brand new concepts can be |During the exploration. At this time, |

| |difficult and intimidating to students |students will work together to discover |

| |individually. So, allowing students to |new terminology, new notation, and new |

| |work in groups will build confidence and |concepts. By working together during this |

| |promote deeper thinking and effort. |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 |During the engagement, explanation, and |

| |learning materials such as their textbook |elaboration. The explanation will use an |

| |and note taking. Using the smart board to |online image projected on the smartboard. |

| |draw or having students use graphing |The explanation will use the smartboard |

| |calculators (when using difficult angle |for the PowerPoint, notes, and examples. |

| |measurements or non-traditional trig |The elaboration will use the smartboard |

| |ratios.) will allow them to explore |where a new problem is projected for |

| |concepts involving trigonometry, degrees, |students to read in front of them and work|

| |radians, and angle measure in a new fun |out a solution. |

| |way. | |

|Questioning in Teaching |Asking questions to the students and |During the exploration, the teacher will |

| |letting them provide the answers requires |ask probing questions to each student. |

| |students to take an active approach to |During the explanation, the teacher will |

| |learning. This is crucial when learning |call on students for their answers from |

| |new topics for the first time, like in |the exploration worksheet. While doing so,|

| |this lesson. |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 |Level 2: Basic Application of |

| |is always increasing or always decreasing allows its inverse to be |Skills & Concepts |

| |constructed. | |

|MACC.912.F-TF.2.7 |Use inverse functions to solve trigonometric equations that arise in modeling |Level 2: Basic Application of |

| |contexts; evaluate the solutions using technology, and interpret them in terms|Skills & Concepts |

| |of the context. | |

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 Functions

Function: Domain: Range:

y = arcsin x iff sin y = x -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2

y = arccos x iff cos y = x -1 ≤ x ≤ 1 0 ≤ y ≤ π

y = arctan x iff tan y = x -∞ < x ≤ ∞ -π/2 < y ≤ π/2

Inverse Properties of Trigonometric Functions

If -1 ≤ x ≤ 1 and -1 ≤ x ≤ 1, then sin(arcsin x) = x and arcsin(sin y) = y

If -1 ≤ x ≤ 1and 0 ≤ y ≤ π, then cos(arcos x) = x and arc(sin y) = y

If 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 Minutes |

|What 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 |Welcome students! My name is Mr. Brenneman / Miss |Good Afternoon! |

|introduce himself/herself and welcome the class. |Walthall. | |

|Teacher will give overview of lesson topic |Today, we will be studying Trigonometric Functions |Oh great, more concepts with trig functions :/ |

| |more in depth, specifically, relating to their | |

| |inverses! |Sounds interesting! |

|Switch to Slide 2 of the Power Point. Teacher will |To begin, I want to show you all an example of some |Why is there a person up on the board?! |

|introduce the engagement activity (Bill Cosby Image.) |math humor. | |

| | |Teacher Response: Because this person helps with the |

| | |logic of this picture. |

|Teachers will ask if any students know who this man |Who knows who this man on the right is? |I don’t know |

|is. | | |

| | |[Bill Cosby] |

|Teachers will ask a question related to this image. |Take a minute and see if you can reason why this |Hmmmm…Something to do with tangent and sine. Where is |

| |equation is true. |the cosine though? |

|Teacher will ask students to explain what their |Who wants to share with the class what they found? |[I will! Basically, sinb/tanb = cosb or “cosby” which |

|reasoning. | |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 |Sounds good! Wonder what’s in store for today’s |

| |start today’s main lesson. |lesson! |

|EXPLORATION 1 Time: 10 Minutes |

|What 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 |What can you tell me about sin (π/2)? |[Sin(π/2) = 1] |

|open with a starting question. | | |

| | |Sin(π/2) = 0. |

|Teachers will ask the reverse of the starting |What if I wanted to go backwards? That is, what angle |[π/2 or 90 degrees] |

|question. |gives us 1 as its sine value? |π/2 or 90 degrees. |

| | | |

| | |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 |So we just found sine inverse of 1. We can denote this|So does this imply: |

|inverse function. |as sin-1(1) or |sin-1(1) = (1/sin)(1) |

| |arcsin(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 |Now let me ask you, what can you tell me about |[Sin(π/4) = [pic]/2] |

|topic. |sin(π/4)? | |

| | |Sin(π/4) = 1 |

|Teachers will ask the reverse of the same question. |What if I wanted to go backwards? That is, what angle |[π/4 and 3π/4] |

| |gives us [pic]/2 as its sine value? (Or what is | |

| |sin-1([pic]/2)?) |π/4 |

| | | |

| | |Teacher Response: Are there any other angles which |

| | |gives us a sine value of [pic]/2? |

|Teachers will elaborate on the previous question. |What could I do to have a unique inverse? That is, |[You can restrict the domain so that we are only |

| |have only one value to represent sin-1([pic]/2), |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 |Notice if I were to restrict the domain to the |What about from π/2 to 3π/2? |

|domain so that the function has a unique inverse. |interval [-π/2, π/2], sinx would take on its full | |

| |range of values, from -1 ≤ sinx ≤1. So in this |Teacher Response: All the sine values in this region |

| |interval, we say sinx has a unique inverse function |have already been accounted for, so we will not be |

| |called the inverse sine function. (Denoted arcsinx or |using this interval when finding the inverse sine |

| |sin-1(x).) |function. |

|Teachers will explain instructions for the Exploration|For this next part, you and your group mates will be |So we are doing similar concepts, but now with the |

|Worksheet. |given additional practice with inverse sine functions,|inverse cosine and inverse tangent function? |

| |as well as exploring inverse cosine and inverse | |

| |tangent functions. Once you have been given a |Teacher Response: Correct |

| |worksheet, you may begin working. If you have a | |

| |question, raise your hand, and we will assist you. | |

|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 |Here is your worksheet. You will have about 5 minute |Misconceptions: |

|with the classmates sitting next to them. Once groups |to complete this. | |

|are made, teachers will pass out the worksheets. | |Students work with their friends on the other side of |

| | |class. |

|EXPLANATION 1 Time: 7 Minutes |

|What the Teacher Will Do |Teacher Directions and Probing/Eliciting Questions |Student Responses and Misconceptions |

|*After 5 Minutes have passed* Teachers will gain |May I have everyone’s attention? We will now be going |Misconceptions: |

|students attention and begin to go over the |over this worksheet. | |

|exploration worksheet. | |Students continue to work on assignment. |

| | |Students continue talking to their classmates. |

|Switch to Slide 4 of the Power Point. Teachers will |Who wants to share what they got for question 1 and 2?|[π/6 and π/4 respectively.] |

|ask what students answered for number 1 and 2. | | |

| | |π/3 and π/2 |

|Teachers will ask students how they came to this |How did you get π/6 and π/4 respectively? |[The interval where sine has an inverse is from –π/2 |

|answer. | |to π/2. Sin-1 (1/2) is π/6 and sin-1 ([pic]/2) = π/4, |

| | |since -π/2 ≤ π/6, π/4 ≤ π/2.] |

|Teachers will ask what students answered for number 3 |Who wants to share what they got for question 3 and 4?|[π/2 and 5π/6 respectively.] |

|and 4. | | |

| | |π/3 and π/2. |

|Teachers will ask students how they came to this |How did you get π/2 and 5π/6 respectively? |[I took a guess and made the interval of the inverse |

|answer. | |cosine function from [0,π]. So cos-1(0) = π/2 and |

| | |cos-1(- |3/2) = 5π/6, since |

| | |0 ≤ 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 |Who wants to share what they got for question 5 and 6?|[0π and –π/4 respectively.] |

|and 6. | | |

| | |π/2 and 7π/4. |

|Teachers will ask students how they came to this |How did you get 0π and –π/4 respectively? |[I took a guess and made the interval of the inverse |

|answer. | |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|We somewhat already learned about the restrictions we |What do you mean when you say the range is: |

|over the definition of the inverse sine function. |need to put on the inverse sine graph so that we have |[-π/2, π/2] and the domain is [-1,1]? |

| |a unique inverse. Notice that the range of the inverse| |

| |sine function is: |Teacher Response: In order for sine to have an inverse|

| |[-π/2, π/2] and the domain is [-1,1]. |function, we must restrict the range from |

| | |[-π/2, π/2]. In this interval, sinx takes on its full |

| |Note: Teachers will go back to the first two questions|range of values |

| |of the exploration worksheet and go over the questions|(-1 ≤ x ≤ 1). |

| |again. | |

|Teachers will go over the definition of the inverse |For the cosine inverse function, we restrict the range|What do you mean when you say the range is [0, π] and |

|cosine function. |to [0, π]. Also notice that the domain of the inverse |the domain is [-1,1]? |

| |cosine function is from [-1,1]. | |

| | |Teacher Response: In order for cosine to have an |

| |Note: Teachers will go back to the next two questions |inverse function, we must restrict the range from |

| |of the exploration worksheet and go over the questions|[0, π]. In this interval, cosx takes on its full range|

| |again. |of values |

| | |(-1 ≤ x ≤ 1). |

|Teachers will go over the definition of the inverse |For the tangent inverse function, we restrict the |What do you mean when you say the range [-π/2, π/2] |

|tangent function. |range to [-π/2, π/2]. Also notice that the domain of |and the domain is [-∞, ∞]? |

| |the inverse tangent function is from | |

| |(-∞,∞). |Teacher Response: In order for tangent to have an |

| | |inverse function, we must restrict the range from |

| |Note: Teachers will go back to the next two questions |[-π/2, π/2]. In this interval, tanx takes on its full |

| |of the exploration worksheet and go over the questions|range of values |

| |again. |(-∞ ≤ 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 |Notice for the sine graph, if I were to restrict the |Wouldn’t the range go from (-∞, ∞) reflecting it over |

|will Switch to Slide 6 of the PowerPoint and break |domain from [-π/2, π/2], it would be one to one (or |the line y = x? |

|down the correlations between the sine, cosine and |pass the Horizontal Line Test.) | |

|tangent graphs with their corresponding inverse | |Teacher Response: Yes it would! But in order to be a |

|graphs. |Reflecting this over the line y = x (which gives us |function, it must pass the Vertical Line Test. If we |

| |the sine inverse function), we see that the domain is |let the graph go to negative and positive infinity, it|

| |restricted to [-1,1] and the range is restricted to |would not pass the Vertical Line Test. |

| |[-π/2, π/2]. | |

|EXPLORATION 2 Time: 5 Minutes |

|What the Teacher Will Do |What the Teacher Will Say (include Probing Questions) |Student Responses and Misconceptions |

|Teachers will present a beginning problem based on |If I were to ask you to evaluate the expression sin |[1/2] |

|inverse properties of trigonometric functions. |[sin-1(1/2)], what would your guess be? | |

| | |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). |You may use the back of your worksheet to answer these|I don’t have any room on the back of my paper. I used |

|Teachers will present 3 more questions about |three questions. Work in your groups. You will have |up all the space. What should I do? |

|properties of trigonometric functions. |about 4 minutes to work on this and afterwards, we | |

| |will go over these problems. |Teacher Response: We have some spare pieces of paper |

| | |you can use. |

|Teachers will gain students attention and begin to |May I have everyone’s attention? We will be going over|Misconceptions: |

|explain the three problems. |these three problems now. | |

| | |Students continue to work on the worksheet. |

| | |Students continue talking with their group mates. |

|EXPLANATION 2 Time: 5 Minutes |

|What the Teacher Will Do |Teacher Directions and Probing/Eliciting Questions |Student Responses and Misconceptions |

|Teachers will ask students what their answers were for|Who wants to share how they evaluated tan [arc tan |[It is just -5 since -5 lies in the domain of the arc |

|the first question. |(-5)]? |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|Who wants to share how they evaluated arc sin [sin |[-π/3 since 5π/3 is coterminal with –π/3.] |

|the second question. |([pic])]? | |

| | |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|Who wants to share how they evaluated cos [cos-1 |[It is not defined since π is not defined.] |

|the third question. |([pic])]? | |

| | |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 |If -1 ≤ x ≤ 1 and – π/2 ≤ y ≤ π/2, then sin(arc sin x)|Misconceptions: |

|provide definitions for inverse properties of |= x and arc sin(sin y) = y. | |

|trigonometric functions. | |Students think the domain and range can be any real |

| |NOTE: Refer to the Power Point for the remaining two |number for the inverse trig functions to apply. |

| |definitions. |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 |What questions do you have regarding these concepts? |Will these inverse properties apply for any arbitrary |

|based on Exploration #2. | |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].] |

|EXPLORATION 3 Time: 5 Minutes |

|What the Teacher Will Do |What the Teacher Will Say (include Probing Questions) |Student Responses and Misconceptions |

|Switch to Slide 9 of the Power Point. Teachers will go|Here, we are given this is a right triangle. We also |[We let tan (32.9) = x/17. Solving for x, we have x = |

|over a question which students should already know how|know one of the angles is 32.9 degrees as well as one |17 tan(32.9) ~ 11. Hence x = 11.] |

|to do. |of the side lengths equaling 17. How would I solve for| |

| |the missing side x? |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 |For this triangle, I am giving you both side lengths. |How are we supposed to solve this if we have never |

|triangle and solve for the angle instead of the side |We know that the angle is going to be the same since |seen this problem before? |

|length. |the side lengths equal the side lengths of the | |

| |previous triangle. Your team’s goal is to try to find |Teacher Response: Use what we did from the previous |

| |a way to get the angle measure. Take a few minutes and|example (where the angle was given), go through the |

| |see if you can figure it out. |steps in the same fashion, and see what you come up |

| | |with. |

| |Time is up. Who wants to share what their group came |[I will! We let tan θ = 11/17. Then we took tan-1 on |

| |up with? |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 Minutes |

|What the Teacher Will Do |What the Teacher Will Say (include Probing Questions) |Student Responses and Misconceptions |

|Teachers will ask students (or groups) if they found a|Time is up. Who wants to share what their group came |[I will! We let tan θ = 11/17. Then we took tan-1 on |

|way to the answer 32.9 degrees. |up with? |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 |What questions do you have regarding this problem? |What if everything was the same except 17 was the |

|regarding this type of problem. | |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 Minutes |

|What the Teacher Will Do |Probing/Eliciting Questions |Student Responses and Misconceptions |

|Switch to Slide 10 of the Power Point. Teachers will |Now, let’s try and use what we have learned to solve a|Can we work in groups? |

|read the problem. |real world problem. | |

| | |Teacher Response: Try and answer this question |

| |“A ladder is placed against a 40 foot high electric |individually. |

| |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. | |

|After 3 minutes have passed, teachers will ask |3 Minutes are up. Who wants to share their answer? |[It is 75.96 degrees] |

|students for their answers. | | |

|Teachers will go over the problem by visually |I am first going to draw a pole which is 40 feet tall.|Misconceptions: |

|representing it on the Smart Board. |From the problem, we know that a ladder is on a tilt | |

| |touching the top of the pole. We also know that base |Why are you using tangent instead of cosine? |

| |of the ladder is 10 feet away from the pole. We set up|Teacher Response: Because if we look at this right |

| |this problem by saying tan θ = 40/10. This implies θ =|triangle and we locate the angle θ, we are given the |

| |tan-1(4) which implies θ = 75.96 degrees. |opposite side and the adjacent side. We use the |

| | |tangent function with this set up. |

|EVALUATION Time: 5 Minutes |

|What 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 |Misconceptions: |

| |work independently. You will have roughly 5 minutes to| |

| |complete this. Once you are finished, raise your hand |Students work in groups. |

| |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([pic]/2) 2) arcsin (1/2)

3) cos-1([pic]) 4) arcos(0)

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

Name: Answer Key

Exploration Worksheet

Evaluate the expression without using a calculator.

1) sin-1([pic]/2) π/4 2) arcsin (1/2) π/6

3) cos-1([pic]) 5π/6 4) arcos(0) π/2

5) tan-1(-1) -π/4 6) arctan (0) 0π

Name: ________________________________

Evaluation

Evaluate the expression without using a calculator.

[pic]

[pic]

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)

[pic]

Name: Answer Key

Evaluation

Evaluate the expression without using a calculator.

[pic] π/3

[pic] 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)

[pic] θ = 48.6°

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