MATH 1523 Instructor Information

Subsubsection Page 3-4

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Instructors are to introduce angles, standard position, positive/negative angles and the two units for measuring angles: degrees and radians.

Ideas to consider:

• Call attention to the name of the angle \(\theta\) (maybe even spell it out) in case some students are not familiar with the Greek alphabet.

• Consider drawing all angles not in standard position until you introduce the definition. Point out that these angles are effectively identical but, at first glance, may not appear so.

• Make a point of identifying the angle when discussing positive and negative angles. Students may only consider the much larger coterminal angle, which may cause confusion later.

• When discussing the two units of angle measure, begin with different units of measurement for other physical quantities (e.g. distance has feet, meters, miles, etc.) and discuss similarities. We will only use degrees and radians for this course.

  • The instructor can give historical context to these units: 360 was chosen by the Babylonians because it was a highly divisible number, \(2\pi\) is related to the circle and arc lengths. The true motivation for radians is explained in Exercise 1.6.

Blanks on page 3:

1. Rays

2. Initial

3. Terminal

4. Standard Position

Subsubsection Exercise 1.1

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The instructor should comment on why the angle for image (c) has a square “arc”. If students struggle with Exercise 1.1, motivate them by considering halving a complete revolution to obtain 180 degrees/\(\pi\) radians etc.

Subsubsection Question

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In general, Questions are intended to spark discussion. There is not always (in fact, rarely) a “correct” answer to the question. These are opportunities to allow students to speak out and come up with creative explanations.

Subsubsection Page 6

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The instructor should write equation (1) and demonstrate deriving the radian measure of 1 degree. The instructor should then show students how to use this fact to convert 15 degrees to radians.

Subsubsection Exercise 1.2

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In Exercise 1.2, the students first try to convert something in radians to degrees using a similar process to above. If they get stuck, ask them to find how many degrees are equivalent to 1 radian.

• For “fun” you can ask them a decimal approximation for this number which should be about 57 degrees.

Then, they are to find an algebraic formula for this process. If they don't know where to begin, ask them to convert theta to degrees. If they get stuck here, ask them to convert \(\pi\) to degrees. Ask them if they can write a formula. If they cannot, ask them to convert some other angle in degrees to radians and repeat the whole process. (This iterative process should end in a finite number of steps.)

Later in Exercise 1.2, the students are asked to do the same for converting angles in degrees to radians. If time is an issue, this and the final part can be assigned for homework that the instructor will go over at the beginning of the next lecture. Alternatively, the lecturer can do part (d) before the students attempt to do Exercise 1.2.

• Parts (c) and (d) should be verified for reasonable answers at some point. These are \dis

Subsubsection Page 9

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The instructor should define coterminal angles using the example. The instructor should show at least two other angles coterminal to \(\dfrac{\pi}{6}\text{,}\) one of which should be a negative angle.

Blank on page 9:

1. Coterminal

Subsubsection Exercise 1.3

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Verify reasonable answers for part (c) (This is \dis).

Subsubsection

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Instructors can remind the class of formulas for areas and circumferences of circles

\(\dfrac{2\pi}{3}\) is one-third of \(2\pi\text{,}\) so that piece of the arc is one-third of the total circumference (which is \(2\pi r\) where \(r\) is the radius of the circle). Therefore, the length of \(s\) is one-third the length of the circumference: \(s = \dfrac{1}{3}(2\pi r)= \dfrac{2\pi r}{3}\)

Blanks on page 11:

1. Arc Length

2. Sector

Subsubsection Exercise 1.4

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Verify reasonable answers for part (d) and part (g). These are \dis

Subsubsection Exercise 1.5

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For Exercise 1.5, make sure to emphasize that if one just blindly used the formula from Exercise 1.4, then one obtains a completely unreasonable number (900 units of length) compared to the circumference (\(24\pi\) units of length). Here, they can make the connection \(s = \dfrac{\theta}{360^\circ} (2\pi r)\) and see that the \(2\pi\)'s do not cancel out.

Subsubsection Question

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Subsubsection Exercise 1.6

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Ultimately, parts (d) and (e) will be used for homework problems and should not be ignored.

Subsubsection Page 16

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The instructor should aim to include as many real-world circular motion scenarios as possible. When discussing the difference between linear speed and angular speed, a useful example is an object in a slingshot. The linear speed of an object in a slingshot is the same before and after it's released (provided there is no friction, of course). The angular speed is really a measure of how many revolutions are happening in a given time frame.

Blanks on page 16:

1. Distance Traveled

2. Arc Length

3. Central Angle, \(\theta\)

4. Rotation in Radians

Subsubsection Exercise 1.7

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For Exercise 1.7, students might need help converting from 15 revolutions every 10 seconds to angular speed. For part (d), students can be told to start with the formula for linear speed and use the hint (and they should conclude linear speed is angular speed times the radius).

Subsubsection Page 19

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Start the worksheet by making the connection between angles and the unit circle. In the same fashion that an angle is a motion, explain that if we rotate an angle all the way around (\(360\dgs\)), then we can obtain a circle. Additionally, there are “pitstops” along the way--our special angles.

• Past students have expressed appreciation if you show them where some of the values on the unit circle came from, perhaps by drawing in an isosceles triangle and using it's properties.

Blanks on pages 19-20:

1. \(\displaystyle x^2+y^2=1\)

2. \(\displaystyle \cos\theta\)

3. \(\displaystyle \sin\theta\)

Subsubsection Exercise 2.1, Exercise 2.2

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Students should be able to use reflections or symmetry to complete the unit circle in Exercise 2.1.

Once they have completed 2.1, they can move directly to Exercise 2.2.

• Many students will try to convert radians to degrees before completing the exercises. Encourage them to begin weaning themselves away from this practice.

• Listen for discussions about negative angles in part (c).

Subsubsection Question

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Make a note of the fact that we usually write \((\sin\theta)^2\) as \(\sin^2\theta\text{,}\) and that this is NOT the same as \(\sin\theta^2\text{.}\) Explain why these are not the same. Do the same for cosine.

Subsubsection Page 24

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Draw a picture on the board or show a video to explain the function as a machine. Be sure to watch their answers on this part of the notes--should be “input” and “output”--NOT “x” and “y”.

Blanks on page 24:

1. Input

2. Output

Subsubsection Exercise 2.3

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You may want to go over this problem as a class to verify everyone has the correct domain and range for part (a), part (b), and part (e). These parts are \dis.

Subsubsection Page 25

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When defining reference angles, be sure to emphasize that these are measured from the “x-axis”, not the “y-axis”. Also, because reference angles are in the interval from \([0,2\pi)\text{,}\) by definition, they are not negative.

Blanks on page 25:

1. 0

2. \(\displaystyle \dfrac{\pi}{2}\)

3. 0

4. \(\displaystyle 90\dgs\)

Subsubsection Exercise 2.4

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You may want to note that an angle that terminates in QI is its own reference angle. Part (d) will require the use of coterminal angles as well as reference angles.

Subsubsection Question

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Subsubsection Exercise 2.5

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This question is asking for three values: the reference angle for \(\theta = -390\dgs\text{,}\) as well as the sine and cosine values for this angle.

Subsubsection Page 28

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Tangent can also be described as a rise (sine) over run (cosine), hinting towards a future definition of a tangent line in Calculus.

Subsubsection Question

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The students can make any ratios they would like. However, the three ratios that define \(\sec\theta\text{,}\) \(\csc\theta\text{,}\) and \(\cot\theta\) will further the discussion toward the next exercise and the definitions that follow.

Subsubsection Exercise 3.1

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Allow students to discuss the problems and work towards the definitions in the following section. Check on groups regularly to guide their progress. It is not necessary to break up the group work between 3.1 and 3.3. The momentum from the last question should help them progress through the next sections, as well. Remember to utilize the Learning Assistant during these longer work sessions.

Subsubsection Page 29

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Verify each group has the proper definitions filled in. While doing so, it will be important to emphasize their domains.

Subsubsection Exercise 3.2

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All or part of this exercise can be sent home for homework.

Subsubsection Question

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Some students may need help determining which value is zero and which value is undefined. Most groups should be able to handle this on their own, but be available to “referee” any discussions that may arise.

Subsubsection Table 3.3

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Groups will finish the tasks between 3.1 and 3.3 at different rates. It is not necessary for every group to finish Table 3.3 before moving on. This can be sent home for homework or used to keep groups busy while other groups “catch up”.

Subsubsection Exercise 3.4

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If necessary, review the definition of a reference angle for part (a). Students should be able to work through the entirety of the exercise before discussing the answers as a class. Letting the groups guide their own way through this exercise is important to learning the uses of reference angles, as well as being able to think through a problem like this on their own at some point. After this, they should be able to see a question similar to part (d) and mentally answer the preliminary parts automatically.

Subsubsection Question

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You may want to facilitate a discussion about mnemonics for remembering the information gained from Exercise 3.4. (“All Students Take Calculus”, “A Smart Trig Class”, etc.)

Subsubsection Page 33

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Draw pictures of \(f(x)=x^2\) and \(f(x)=x^3\) to illustrate typical properties of even and odd functions. Then, you can draw a unit circle and review the definition of a negative angle. This topic presents opportunity to reinforce discussions about input values and output values.

Subsubsection Exercise 3.5

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If necessary, remind students about reference angles to find the negative inputs. Instruct the groups to move on to the next section when you see most of them have completed Table 3.6.

Subsubsection Table 3.6

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Subsubsection Exercise 3.7

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As you walk around, you may notice students initially guessing \(\dfrac{\pi}{6}\) for the period of sine.

Subsubsection Exercise 3.8

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Let the groups work on this section together. Make sure the groups can explain why the period of the tangent function is not \(2\pi\text{.}\)

Subsubsection Exercise 3.9

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This exercise makes use of many definitions learned over the last three worksheets including reference angles, coterminal angles, and negative angles.

Subsubsection Table 3.10

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Subsubsection Exercise 3.11

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If time is short, this exercise can be sent home for homework. Verify students use a correct identity for part (d). This is \disc.

Subsubsection Exercise 3.12

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Keep watch out for groups that find the other two Pythagorean Identities. Allow a representative from the group(s) to present the new identities. If no group has completed them, start with the most progressed ideas and guide the class towards the final results.

Subsubsection Page 40-41

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This is where we make the connection between unit circle trig definitions and right triangle trig definitions. Draw the unit circle and a general angle (it doesn't have to terminate in QI but be ready to discuss reference angles and acute angles, if necessary) and then drop perpendiculars from the point of intersection. Now we can concentrate our study on the triangle that results.

Blanks on page 40-41:

1. 1

2. \(\displaystyle \cos\theta\)

3. \(\displaystyle \sin\theta\)

Subsubsection Question

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We are hoping to make the connection between the radius of the circle and the denominator so we can scale!

Subsubsection Exercise 4.1

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Yo! This exercise is harder than it looks! DON'T WING IT!! The point of this exercise is to connect the unit circle to right triangle trig.

Blanks for part (c):

1. hypotenuse

2. right

3. opposite/hypotenuse or O/H

4. adjacent/hypotenuse or A/H

5. opposite/adjacent or O/A

6. SOH CAH TOA

Using similar triangles, set up the ratios \(\dfrac{\cos\theta}{1}=\dfrac{x}{5}\) and \(\dfrac{\sin\theta}{1}=\dfrac{y}{5}\) for parts (e) and (f).

The answer to the question asked in part (g) is “the intersection point of this larger circle with the terminal side of \(\theta\)” or some such thing.

Blank in Note:

1. positive

Subsubsection Exercise 4.2

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This exercise is relating the Pythagorean Identity \(\sin ^2\theta + \cos ^2\theta = 1\) to the Pythagorean Theorem \(a^2 + b^2 = c^2\) that students are familiar with. Because they are so familiar with the Pythagorean Theorem, they can stumble over the algebra required in this exercise.

Subsubsection Exercise 4.3

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If you send Exercise 4.2 home for homework, you may need to prompt students to recall the Pythagorean Theorem to complete part (b).

Subsubsection Question

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The students can answer this question either way, as long as they justify their response.

Subsubsection Exercise 4.4

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For part (d), the students will either use the Pythagorean Theorem or they will use a different trig function. Ask them to use the other process for part (e). This is \dis.

Subsubsection Exercise 4.5

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Blank in Note:

1. \(90\dgs\) or \(\dfrac{\pi}{2}\)

The students will struggle the most with part (d). This is \dis. If necessary, do this as a class. Make sure they write the formulas in both degrees and radians.

For part (e), discourage the students from converting to degrees for the second set of angles.

Subsubsection Exercise 4.6

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The students may label the wrong angle with \(\theta\text{.}\) Verify everyone is using the correct angle when defining the Angle of Depression/Elevation.

Subsubsection Exercise 4.7

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Ask students to pose an Angle of Elevation problem. Make sure they include all needed information in their problem posing!

Subsubsection Page 54

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The instructor should begin by introducing the ant example, and then turn the groups loose and only call them back together for the second question on page 58.

Blanks on pages 54-56:

1. a few points

2. curve

3. sine

4. basic sinusoid

5. sinusoid

Subsubsection Page 55-58

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To plot \(f(\theta)\) and \(g(\theta)\text{,}\) they should plot the specified points and fill in a reasonable curve. The instructor should go around and individually show what a sinusoid looks like and tell them the names of these shapes. When they want to extend these functions to bigger domains, they need to recall that these functions are periodic. In the context of the ant example, it should be clear that the pattern repeats.

Subsubsection Question

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For the comparison question, the students should conclude: the shapes are essentially same and that cosine is sine shifted over.

Subsubsection Question

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For the information question, make them write something (anything) down and then bring them back to this question at the end.

Possible answers:

You may want to explain the 5-point method: students should graph the basic sinusoid first and then extend it to a full sinusoid. The student should specify the 5 points displayed on each basic sinusoid.

Subsubsection Page 60

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You could use this space for discussing the general transformations of functions that the students have seen in other classes.

Subsubsection Exercise 5.1

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Subsubsection Exercise 5.2

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Subsubsection Definition

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Amplitude

If necessary, point out that amplitude is not always equal to the maximum output value.

Subsubsection Exercise 5.3

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The students are reminded about reflections of graphs in part (a).

Part (b) addresses the absolute value portion of the definition of amplitude.

Part (d) is a nice visualization question. This is \disc.

Subsubsection Exercise 5.4

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The tables in part (a) are formatted differently than many students are used to. The output values are given in the top row. The second row is intended as space to record intermediate calculations with a variable substitution (the notation is not perfect but it suffices.) The last row of \(t\) values are needed for the graph in part (b). It is common for students to mistakenly use coterminal angles rather than use angles greater than \(2\pi\) after the first rotation.

Subsubsection Exercise 5.4

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1. time

2. oscillation

Subsubsection Exercise 5.5

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Subsubsection Exercise 5.6

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This exercise introduces phase shifts to the students in terms of the ant and wheel example. For Exercise 5.6 part (d), if they are struggling, ask them to think about how much further ahead the first ant is compared to the other one. When they conclude it's a fixed amount, ask them to make an algebraic expression out of \(t\) so that the angle of the two ants correspond to the same \(t\text{.}\)

Subsubsection Exercise 5.7

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This is a concrete example of a phase shift, i.e. we realize cosine as a phase shift of sine. The students might pick different points for the tables in part (a), but the graphs should be identical.

Make sure to point out the effect period has on the phase shift. For example, \(f(x) = \sin(B x - C)\) would have a phase shift of \(\dfrac{C}{B}\text{,}\) but \(g(x) = \sin(B(x - C))\) would have a phase shift of \(C\text{.}\)

Subsubsection Exercise 5.8

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The groups should be able to do Exercise 5.8 entirely on their own. By this point in the worksheet, they usually feel more comfortable with the notation.

Subsubsection Question

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One possible answer: think about moving the wheel around on the coordinate plane and having functions that measure the distance from the ant to the horizontal and vertical axis. If groups struggle to arrive at this, then show them how to obtain \(\sin(\theta) + 5\) and let them figure out the rest.

Subsubsection Exercise 5.9

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The students do not need to graph these functions. However, once they have completed Exercise 5.10, you could ask them to do so for homework.

Subsubsection Exercise 5.10

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Subsubsection Exercise 5.11

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All four transformations are present in the equation for the height function.

Subsubsection Question

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There are infinitely many ways to write this function but you could ask the students to find one using the “other function” they didn't use in part (d).

Subsubsection Table 6.1

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If necessary, write the previous definitions of tangent on the board with the assistance of the students.

Subsubsection Note

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Blank in Note:

1. asymptote

Subsubsection Exercise 6.2

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Observe student interaction for the following problems. Ideally, they will be able to move through this section quickly after seeing sine and cosine graphs previously. Issues may arise when they have to decide what is happening when the function is undefined. Encourage groups to plug in additional values surrounding the undefined value and deduce that an asymptote is necessary.

Subsubsection Exercise 6.3

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Again, ask the students for the previous definition of cotangent and write it on the board for a small review.

Subsubsection Exercise 6.4

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Review the definitions of secant and cosecant as ratios of sine and cosine.

Subsubsection Exercise 6.5

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Many students may refer to this kind of transformation using the word 'amplitude.' You might remind them that the tangent graph can't satisfy the definition of amplitude. However, should you choose to have a 'class dialect' that refers to vertical stretch this way, make sure you are pedantic about it at least once!

Subsubsection Exercise 6.6

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This exercise is similar to Exercise 5.4 in the previous worksheet and is concerned with the period of tangent.

Subsubsection Exercise 6.7

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As with the exercises in Worksheet 5, phase shift and vertical shift follow after vertical and horizontal stretching. This exercise combines both of the last transformations.

Subsubsection Exercise 6.8

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This exercise sums up all of the transformation content covered in Worksheet 5 and 6. It is the first time they are asked to consider multiple transformations applied to the same (new) function. If you send it home for homework, confirm that students are arriving at a correct or reasonable answer and have a discussion in class as necessary.

Subsubsection Question

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Possible answers include:

• intercepts

• asymptotes

• maximums/minimums

• period

• basic shape

Subsubsection Exercise 6.9

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Exercise 6.9 should be done as a group in class. Students will be combining these transformations graphically for the first time, and it has proven to be one of the more challenging topics of the semester. They will need positive feedback and guidance as they try these out.

Subsubsection Exercise 6.10

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Each of these has some combination of transformations for students to graph. Students will need to pay attention to labels as they draw these graphs.

Subsubsection Question

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Let the students come up with functions and the inverses. Some of them may remember how to “find” inverse functions. \(x^2\) is a very useful function.

Subsubsection Exercise 7.1

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If necessary, clarify the definition of inverse and inform the necessity of the reciprocal functions secant, cosecant, and cotangent. The notation for an element, \(x^{-1}\text{,}\) is in reference to the multiplicative inverse of that element. However, an inverse function is not to be confused with a multiplicative inverse. There should never be confusion about whether the negative one on a trig function means an inverse or reciprocal.

It is also helpful here to discuss an analogy for inverse functions. You put on your socks before you put on your shoes, so to invert this action, you simply must remove your shoes before removing your socks. An inverse is to be considered the opposite of the original action. This should also help clarify the issue of reciprocal vs. inverse. An inverse of a multiple is a reciprocal, but an inverse of a sine function is an arcsine function.

Let the students argue about parts (e) and (f). If necessary, clarify for future use that the inverse trigonometric functions are sometimes referred to as arcsine, arccosine, and arctangent.

Subsubsection Exercise 7.2

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The students should be able to do this problem with very little assistance. If necessary, discuss part (d). This is \disc.

Subsubsection Exercise 7.3

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The students should draw two periods of sine, cosine, and tangent. It is important the students draw two periods for tangent or the “inverse” will appear as a function.

Subsubsection Exercise 7.4

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This is a good place for a discussion about one-to-one functions and why they are necessary for inverses. The students can identify any interval that would make each function one-to-one. This will help with solving in later exercises.

Subsubsection Table 7.5

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Give the students the conventional restricted domains for sine, cosine, and tangent.

Subsubsection Question

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The students can find any number of restricted domains that satisfy the various conditions. This question is an important lead-in to the next exercise.

Subsubsection Exercise 7.6

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The following problems are to practice finding inverse values on various domains. Give them approximately 5 minutes, but if they don't finish, these are good problems to practice at home. These could be \hw.

Subsubsection Exercise 7.7

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Allow students to finish the following. After approximately 10 minutes, for parts (d) or (e) (these are \disc), ask a group that has finished or has gotten furthest to explain the exercises. Guide them through the solution with some leading questions.

Subsubsection Questions

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Have an open discussion in the class about these two questions. THROUGHOUT this worksheet, make sure students understand that we cannot just assume these “=” to be true, but that we are proving it is or is not true. Make sure they know the difference between setting two things equal and then altering both sides(which we do NOT want to do) and showing that one side of an equation actually equals the other(which we DO want to do)!

Subsubsection Question

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This question should be inspiration for either creating a formula reference or filling out the appendix in the back of the workbook. Time permitting, go around the room and ask students what they thought was important.

Subsubsection Exercise 8.1

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Show the students the difference between solving and isolating terms and verifying identities. We cannot assume the properties of equality so we cannot add, subtract, multiply, or divide across the equal sign. One technique for verification is rewriting expressions in terms of sine and cosine.

Part (c) and (d) illustrate that these identities only hold on the domains of the functions involved. Therefore, the adjustment to the definition of verification, should include language about domain.

Subsubsection Exercise 8.2

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For the quiz and exam, students may be asked to justify each step in their verification. Algebra and simplification are not as emphasized as trigonometric definitions or identities used.

Subsubsection Exercise 8.3

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This is a good candidate for \hw.

Subsubsection Note

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Explain the meaning of substitution, and how it is helpful.

Subsubsection Exercise 8.4

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Subsubsection Exercise 8.5

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This is a good candidate for \hw.

Subsubsection Exercise 8.6

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This exercise does not give the students the “other side” of the identity. They will use two versions of Pythagorean Identities to rewrite and simplify the expression. Hopefully, they will see that both methods lead to the same result, showing that there is not one “RIGHT” way to approach the problem.

Be careful, many students will not read the instructions and will try to put all expressions in terms of sine and cosine.

Subsubsection Exercise 8.7

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Subsubsection Exercise 8.8

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Sometimes, the students become complacent when every problem can be verified so Exercise 8.8 will fail. However, they are then asked to “adjust” the expressions in three distinct ways to make it an identity.

This is a good candidate for \hw.

Subsubsection Question

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Groups are to come up with whatever they want for the first question. Likely, they will think sine is additive (i.e. \(\sin(x + y) = \sin(x) + \sin(y)\)). They will then test their formula for some special angles, and see that there is a problem.

Subsubsection Exercise 9.1

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Subsubsection A Formula Appears

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The instructor gives them the non-intuitive answer and they complete the Exercise 9.1.

\(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \sin\beta\cos\alpha\)

Subsubsection Exercise 9.2

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For Exercise 9.2, groups only need to note that

to apply the sum formula. For part (d), instructors might need to remind the class about even and odd functions.

Subsubsection A Formula Appears

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\(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \sin\beta\cos\alpha\)

Subsubsection Exercise 9.3

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Subsubsection Exercise 9.4

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Exercise 9.4 is similar to Exercise 9.2.

Subsubsection Exercise 9.5

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Prepare to spend 20 minutes on Exercise 9.5.

Groups will have to think a bit about part (b). The trick is to divide the numerator and denominator by \(\cos(\alpha)\cos(\beta)\text{.}\) But, this might take a bit to see. Students will likely make a mistake or two with fractions. It's also possible they end up with the things in terms of cotangent which might add some extra time. Give them the chance to explore and discuss.

Subsubsection Exercise 10.1

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This exercise scaffolds the derivation of the three double angle identities for cosine. The most common mistake students make in this and following exercises is leaving the left-hand side of the equation out of their notes. It becomes a problem later for other proofs.

Subsubsection Exercise 10.2

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Similar to the exercise above, but Sine and Tangent only have one double angle identity each so this should be much faster than Exercise 10.1.

Subsubsection Exercise 10.3

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This exercise might require a bit of help with drawing the triangle in part (a). If necessary, have a class discussion about part (d) and how signs are determined.

Subsubsection Exercise 10.4

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This should be a straight-forward exercise for the students to do in groups.

Subsubsection Exercise 10.5

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This exercise might require a bit more intervention from you. If needed, help the students make the appropriate substitution for part (a).

Subsubsection Angles and Half-Angles

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This is where most of your lecture will take place. Discuss how half-angles are created by halving each of the quadrants in turn. For this course, we will only assume half-angles are created from one revolution around the unit circle.

Part (e) is a good \hw problem.

Subsubsection Exercise 10.6

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Subsubsection Exercise 10.7

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Once Exercise 10.5 is complete, students should be able to complete Exercise 10.6 and 10.7 by following a similar procedure.

Subsubsection Exercise 10.8

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Like Exercise 10.3, students might need a little more help from you on this one. Give them some time to try it out before you call them together.

Subsubsection Exercise 10.9

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Now that the students have new identities, more complicated proofs can be verified.

For part (c), start with the left side even though the right side is “uglier”.

Subsubsection Question

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Subsubsection Comment

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Blank on page 145:

1. domain

Subsubsection Question

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There are not any specific “correct” answers for this question. Students may list any restricted domain that meets the given criteria.

Subsubsection Exercise 11.1

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This exercise asks the students to think about things more geometrically. The point of this question is to get students to draw straight lines to help them identify potential points that solve their trig equation.

Subsubsection Exercise 11.2

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Subsubsection Exercise 11.3

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They are mostly review of the inverse trig section, so they might take less time given how well the students remember that section. The overall strategy they should develop is to first find all the solutions in the typical domain $[0,2\pi)$, and then translate the solutions to the appropriate domain.

If needed, send one of these for \hw

Subsubsection Exercise 11.4

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Subsubsection Exercise 11.5

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Subsubsection Exercise 11.6

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These problems are intended to expand on the \(y=mx+b\) equation. Some will require square roots, reciprocals, or combining like terms. Instructors should emphasize that students double check their answers.

Subsubsection Exercise 11.7

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This equation is different because it contains two distinct trig functions instead of one. It will require some different strategies than have been used so far.

Subsubsection Exercise 11.8

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Subsubsection Exercise 11.9

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Subsubsection Exercise 11.10

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Exercises 11.8 and 11.9 will factor “nicely”. 11.10 will not...

The main issue you may find is that students will stop once they have an expression like \(\sin\theta = \dfrac{1}{2}\text{.}\) Encourage them to keep going until they find solutions for \(\theta\text{.}\)

Subsubsection Exercise 11.11

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In this exercise, students will use an identity to change the double angle portion into quadratic-type terms.

Subsubsection Exercise 11.12

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The first asks them to manipulate the domain and solve using a substitution.

Subsubsection Exercise 11.13

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The second asks them to make a substitution to find the initial solutions, and then add the period to find the entire solution set.

Subsubsection Exercise 11.14

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Multiple angle equations will only appear on the exam with multiples of three or more, to keep students from being confused about which strategy to employ.

Subsubsection Exercise 11.15

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This problem is very similar to the Ferris Wheel problem in Worksheet 5. Students should have experience with creating the function that represents the scenario, the rest of the exercise should be a straight-forward extension of this worksheet. However, be prepared to help students find the second solution using reference angles, if they get stuck!

Each of the following problems are application problems. To be prepared for hard questions from the students, instructors might want to research what these concepts are if they are unfamiliar. For the second exercise in this section, the equilibrium position just means that the displacement \(p(t)\) is zero.

Exercises 11.16-11.18 are optional.

Subsubsection Page 166

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Blanks on page 166:

1. oblique

2. altitude

3. perpendicular

Subsubsection Exercise 12.1

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They may need assistance with clarifying part (c). Once they are able to do this part, though, the following parts should be clear. After about 5 minutes, ask a group to explain what they have so far for part (c) and have their classmates assist in finishing the proof, if necessary. This should help the class with the following parts and save time.

Subsubsection Exercise 12.2

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The following problem is a clear cut example of how to use Law of Sines. We start with the non-ambiguous case to illustrate the ease of using Law of Sines.

Subsubsection Exercise 12.3

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Now they will have to learn about ambiguity. This example will take longer because it may not be immediately apparent that they have been given an acute angle for what is pictured to be obtuse. If it is immediately clear to them, they may not see that this means there will be two possible triangles, but rather that the picture is incorrect.

Subsubsection Exercise 12.4

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This exercise will introduce the scenario in which no triangle satisfies the initial conditions.

Subsubsection Exercise 12.5

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These problems are for practicing the ideas introduced in Exercise 12.2-12.4. If necessary, this problem can be sent home as \hw

Subsubsection Exercise 12.6

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To help the students derive the area formula, have them first consider how to find the area of a parallelogram (\(A=base\cdot height\)), then observe that an oblique triangle is half of a parallelogram. Then, make sure they can find the height in terms of the sides and angles given, so that their final formula is not in terms of height.

Subsubsection Exercise 12.6

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This question is an opportunity to practice using the area formula.

Subsubsection Exercise 12.8

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Students may need help with drawing a picture for exercise 12.8, but if time permits, let them figure it out on their own. Having an instinct for drawing figures in applications is something they need to start building.