TRIGONOMETRY PACKET

# 11 - 16

Hamby

Trigonometry Packet #11 page 1 of 2

Use the identities above to rearrange the equations for the given term:

1. 𝐅𝐢𝐧𝐝 𝐬𝐢𝐧𝟐𝛉 when 𝐬𝐢𝐧𝟐𝛉 + 𝐜𝐨𝐬𝟐𝛉 = 𝟏. (in other words, get 𝐬𝐢𝐧𝟐𝛉 by itself)

2. 𝐅𝐢𝐧𝐝 𝐜𝐨𝐬𝟐𝛉 when 𝐬𝐢𝐧𝟐𝛉 + 𝐜𝐨𝐬𝟐𝛉 = 𝟏. (in other words, get 𝐜𝐨𝐬𝟐𝛉 by itself)

3. 𝐅𝐢𝐧𝐝 𝐭𝐚𝐧𝟐𝛉 when 𝐭𝐚𝐧𝟐𝛉 + 𝟏 = 𝐬𝐞𝐜𝟐𝛉.

4. Find 𝟏 𝐰𝐡𝐞𝐧 𝐭𝐚𝐧𝟐𝛉 + 𝟏 = 𝐬𝐞𝐜𝟐𝛉.

5. Find 𝐜𝐨𝐭𝟐𝛉 𝐰𝐡𝐞𝐧 𝟏 + 𝐜𝐨𝐭𝟐𝛉 = 𝐜𝐬𝐜𝟐𝛉.

6. Find 𝟏 𝐰𝐡𝐞𝐧 𝟏 + 𝐜𝐨𝐭𝟐𝛉 = 𝐜𝐬𝐜𝟐𝛉.

Keep these handy, you will use these on the proving identities problems.

Trigonometry Packet #11 page 2 of 2

In each, prove that the left side is equal to the right side using the trig identities.

1. −𝐭𝐚𝐧𝛉𝐜𝐨𝐬𝛉 = −𝐬𝐢𝐧𝛉 2. 𝐬𝐞𝐜𝟐𝛉 − 𝟏 = 𝐭𝐚𝐧𝟐𝛉

3. 𝐬𝐞𝐜𝛉

𝐜𝐬𝐜𝛉 = 𝒕𝒂𝒏𝟐𝜽 𝟒. 𝐜𝐬𝐜𝟐𝛉 − 𝐜𝐨𝐭𝟐𝛉 + 𝐬𝐢𝐧𝟐𝛉 = 𝟏 + 𝐬𝐢𝐧𝟐𝛉

5. 𝐬𝐞𝐜𝛉𝐜𝐨𝐭𝛉𝐬𝐢𝐧𝛉 = 𝟏 6. 𝐜𝐬𝐜𝛉𝐜𝐨𝐬𝛉𝐭𝐚𝐧𝛉 = 𝟏

8. 𝟏+𝐭𝐚𝐧𝟐𝛉

𝟏+𝐜𝐨𝐭𝟐𝛉= 7. 𝒄𝒐𝒕𝟐𝜽(𝟏 + 𝒕𝒂𝒏𝟐𝜽)

Hint: substitute

using a

Pythagorean

identity.

Substitute a

quotient identity

for 𝑐𝑜𝑡2𝜃 and

reciprocal

identity for

𝑠𝑒𝑐2𝜃. Cross-

cancel.

Reciprocal Identities

Hint: substitute

using a 2

Pythagorean

identities.

Substitute 2

reciprocal

identities. Keep

change flip.

Substitute using

a quotient

identity.

Trigonometry Packet #12 page 1 of 2

Example 1:

You try a few like this:

1. 𝒔𝒊𝒏𝜽

𝟏+𝒄𝒐𝒔𝜽=

𝟏−𝒄𝒐𝒔𝜽

𝒔𝒊𝒏𝜽 2.

𝒄𝒐𝒔𝟐𝜽

𝟏−𝒔𝒊𝒏𝜽= 𝟏 + 𝒔𝒊𝒏𝜽

Hints for Verifying Trig Identities:

1. Memorize the Reciprocal Identities, Quotient Identities and Pythagorean Identities.

2. If there is a squared term, such as 1 − cos2θ, consider if a pythagorean identity could be substituted.

3. Try to re-write the most complicated side.

4. Sometimes it is helpful to turn all of the functions into sines and cosines by substituting an identity.

5. If there are 3 terms, can it be factored? If there is a difference of 2 squares, can you factor?

6. Sometimes, it may help to simplify both sides of the equation, then try to get them to match.

7. If an expression contains a 1 + sin 𝜃 or 1 + 𝑐𝑜𝑠𝜃, try to multiply BOTH the numerator and denominator

by the conjugate (1 − sin 𝜃) 𝑜𝑟 (1 − cos 𝜃) respectively.

8. Consider getting a common denominator. Or, when you have a common denominator with 2 terms in

the numerator, split up the fraction into 2 fractions.

9. You always MUST NEVER cross sides of the equation. Always keep the sides separate.

cosθ

1−sinθ=

1+sinθ

cosθ I look at this, and I see each side looks equally complicated. Everything is

already in sines and cosines. The only thing I notice is that the denominator is 1 − 𝑠𝑖𝑛𝜃. We can use

hint #7 above on this problem.

On the left, multiply the numerator and denominator by the conjugate of 1 − 𝑠𝑖𝑛𝜃 which is 1 + 𝑠𝑖𝑛𝜃.

Then FOIL the denominator.

cosθ

1−sinθ∙

1+𝑠𝑖𝑛𝜃

1+𝑠𝑖𝑛𝜃=

𝑐𝑜𝑠𝜃(1+𝑠𝑖𝑛𝜃)

1−𝑠𝑖𝑛2𝜃 . We can then use a Pythagorean identity on the denominator.

𝑐𝑜𝑠𝜃(1+𝑠𝑖𝑛𝜃)

𝑐𝑜𝑠2𝜃. Now, we can cancel one cosine. We have left:

1+𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃 which is what we were trying

to prove.

Trigonometry Packet #12 page 2 of 2

EXAMPLE 2:

You try a few!

3. 𝟏

𝒔𝒊𝒏𝜽+

𝟏

𝒄𝒐𝒔𝜽=

𝒔𝒊𝒏𝜽+𝒄𝒐𝒔𝜽

𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽 4.

𝒔𝒊𝒏𝜽

𝒄𝒐𝒔𝜽+

𝒄𝒐𝒔𝜽

𝒔𝒊𝒏𝜽=

𝟏

𝒄𝒐𝒔𝜽𝒔𝒊𝒏𝜽

cotθ + tanθ = secθcscθ. Seems like the left side might be more complex (hint 3). We can also reduce that left

side using sines and cosines (hint 4).

Left side: cosθ

sinθ+

sinθ

cosθ. Now, we need to use hint 8 to get a common denominator. Side note, when you get

a common denominator, you have to multiply the numerator and denominator by the missing factor, like this: 2

3+

4

5=

2

3∙

5

5 +

4

5∙

3

3 =

10

15+

12

15 =

22

15 We need to use this same process in our trig problem. We can

use the common denominator of 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃.

cosθ

sinθ∙

𝑐𝑜𝑠𝜃

𝑐𝑜𝑠𝜃+

sinθ

cosθ∙

sinθ

sinθ. This makes

𝑐𝑜𝑠2𝜃

sinθ𝑐𝑜𝑠𝜃+

𝑠𝑖𝑛2𝜃

sinθ𝑐𝑜𝑠𝜃. When we add them we get

𝑐𝑜𝑠2𝜃+𝑠𝑖𝑛2𝜃

sinθ𝑐𝑜𝑠𝜃.

We use a Pythagorean identity to get 1

sinθ𝑐𝑜𝑠𝜃, which is

1

sinθ∙

1

𝑐𝑜𝑠𝜃= cscθ ∙ secθ or secθ ∙ cscθ which is

what we were trying to prove!

Trigonometry Packet #13 page 1 of 4

Multiply Each, using distribution or FOIL.

1. (𝒄𝒐𝒔𝜽 + 𝟐)(𝒄𝒐𝒔𝜽 − 𝟓) 2. (𝒔𝒊𝒏𝜽 + 𝟑)(𝒔𝒊𝒏𝜽 − 𝟑)

3. (𝒄𝒐𝒔𝜽 + 𝒔𝒊𝒏𝜽)(𝒄𝒐𝒔𝜽 + 𝒔𝒊𝒏𝜽) 4. (𝟐𝒄𝒔𝒄𝜽 − 𝟏)(𝟑𝒄𝒔𝒄𝜽 − 𝟖)

More Trig Factoring… Look back at packet 10 for hints

5. 𝒄𝒐𝒕𝟐𝜽 − 𝟏 6. (𝒔𝒊𝒏𝜽 + 𝟏)(𝒔𝒊𝒏𝜽 + 𝟏)

7. 𝟑𝒄𝒐𝒔𝟐𝜽 + 𝟕𝒄𝒐𝒔𝜽 + 𝟐 8. 𝟒𝒄𝒔𝒄𝟐𝜽 + 𝒄𝒔𝒄𝜽 − 𝟑

9. 𝒄𝒐𝒔𝟒𝜽 + 𝟐𝒄𝒐𝒔𝟐𝜽 + 𝟏 10. 𝒄𝒐𝒔𝟑𝜽 + 𝒔𝒊𝒏𝟑𝜽 (see sum of cubes formula)

Trigonometry Packet #13 page 2 of 4

West Virginia Data from http://covid19.healthdata.org/ as of 3.31.2020

1. When is the peak number of deaths per day predicted?

2. The pink shade indicates a range of possible deaths per day. On May 1, what is the range number of deaths

predicted for that day? June 1 range?

3. If we return to school on April 20, approximately how far will we be through this covid-19 cycle?

4. From beginning to end, how long does this project the covid-19 crisis to last?

5. When is our highest rate of increase in deaths (date range)?

6. What function type does this data most represent?

7. Over what interval are the deaths increasing? Decreasing?

8. Is the function concave up or concave down?

9. What is the domain and range of the dashed line prediction?

10. Based on this data, when do you believe the Governor should life the stay-at-home order?

Trigonometry Packet #13 page 3 of 4

Total Deaths

11. How many total deaths are projected for WV?

12. What is the possible range of total deaths by August 1?

13. By May 1, what are the total projected deaths?

14. When do we project the to reach the peak of total deaths?

15. Why does the curve level off after mid-June?

16. Will the curve decrease, like a quadratic model would? Why or why not?

17. What are interventions that could occur that would change these death predictions? List as many as you can.

18. If you had to advise Jim Justice of when it was safe to lift the stay-at-home order based on this data, what

would you suggest?

Trigonometry Packet #13 page 4 of 4

Hospital Resources in WV

19. How many beds are available in the State? How many beds, maximum, do they predict we need?

20. How many ICU beds do we have available, approximately? How many do they predict we may need by

May 1? If the prediction were true, on May 1 would we have enough ICU beds?

21. Give a summary on our supplies for covid-19 in the state. Are we prepared with beds, ICU beds, and

ventilators?

22. Can we use this graph to give Jim Justice an end date for the stay-at-home order? Why or why not?

Trigonometry Packet #14 page 1 of 2

Review of adding fractions with like denominators:

Add the fractions: Don’t use your calculator!

1. 𝟏

𝟑+

𝟓

𝟐 2.

𝟕

𝟑+

𝟏

𝟒

3. 𝟓

𝟔+

𝟑

𝟒 4.

𝟏

𝟔+

𝟓

𝟑

Now, add these. Again, no calculator. This applies to our upcoming lesson.

𝑬𝒙𝒂𝒎𝒑𝒍𝒆: 𝝅

𝟑+

𝝅

𝟐=

𝟐𝝅

𝟔+

𝟑𝝅

𝟔=

𝟓𝝅

𝟔

5. 𝟐𝝅

𝟑+

𝟕𝝅

𝟒= 6.

𝟏𝟏𝝅

𝟔+

𝟑𝝅

𝟒=

7. 𝟓𝝅

𝟑+

𝟓𝝅

𝟔= 8.

𝝅

𝟒+

𝟐𝝅

𝟑=

25

40+

32

40+

28

40=

85

40

Trigonometry Packet #14 page 2 of 2

Create two unit circle fractions (radian angles) that add up to each sum.

This will take some trial and error, and re-trying! This applies to our upcoming lesson.

Example: ____+____ = 𝟓𝝅

𝟔 So

𝝅

𝟑+

𝝅

𝟐=

𝟓𝝅

𝟔

So, you should work backwards to come up with the 2 fractions.

9. ____ + ____ =𝟓𝝅

𝟏𝟐 10. ____ + ____ =

𝟕𝝅

𝟏𝟐

11. ____ + ____ =𝟏𝟗𝝅

𝟏𝟐 12. ____ + ____ =

𝟑𝟓𝝅

𝟏𝟐

13. ____ + ____ =𝟑𝝅

𝟐 14. ____ + ____ =

𝟓𝝅

𝟐

15. ____ + ____ =𝟏𝟎𝝅

𝟑 16. ____ + ____ =

𝟏𝟗𝝅

𝟔

Trigonometry Packet #15 page 1 of 6 Read/Analyze pages 1-3

Opinion: Modern high school math should be about data science — not Algebra 2

(Patrick T. Fallon / For The Times) By JO BOALER , STEVEN D. LEVITT; OCT. 23, 2019; 3 AM

Thanks to the information revolution, a stunning 90% of the data created by humanity has been generated in just the past two years. Yet the math taught in U.S. schools hasn’t materially changed since Sputnik was sent into orbit in the late 1950s. Our high school students are taught algebra, geometry, a second year of algebra, and calculus (for the most advanced students) because Eisenhower-era policymakers believed this curriculum would produce the best rocket scientists to work on projects during the Cold War. It has been 50 years since the U.S. reached the moon, almost 30 years since the Berlin Wall fell. Technology has advanced to the point that tiny powerful computers are routinely carried around in pockets and purses. Times have changed, and so has the math people use in everyday life. We surveyed 900 “Freakonomics” podcast listeners — a pretty nerdy group, we must admit — and discovered that less than 12% used any algebra, trigonometry or calculus in their daily lives. Only 2% use integrals or derivatives, the foundational building blocks of calculus. In contrast, a whopping 66% work with basic analytical software like Microsoft Excel on a daily basis. When was the last time you divided a polynomial? If you were asked to do so today, would you remember how? For the most part, students are no longer taught to write cursive, how to use a slide rule, or any number of things that were once useful in everyday life. Let’s put working out polynomial division using pencil and paper on the same ash heap as sock darning and shorthand. What we propose is as obvious as it is radical: to put data and its analysis at the center of high school mathematics. Every high school student should graduate with an understanding of data, spreadsheets, and the difference between correlation and causality. Moreover, teaching students to make data-based arguments will endow them with many of the same critical-thinking skills they are learning today through algebraic proofs, but also give them more practical skills for navigating our newly data-rich world. Data-based math courses allow students to grapple with real-life problems. They might analyze issues about the environment, space travel or nutrition. Students can examine the threat of wildfires or the ways social media is tracking their data, learning how to apply math to real-world issues. Other countries are moving much faster than the U.S. in instituting such a curriculum. Over the last 50 years, statistics and data science have become an integral part of the United Kingdom curriculum. Canada’s educational system, which is ranked highly internationally, also incorporates statistics and data. In addition, the Program for International Student Assessment, or PISA, measures how effectively countries are preparing students for the mathematical demands of the 21st century. Last week, PISA released a mathematics framework that guides the assessments. Data literacy is central to the framework. In contrast, U.S. high school students learn algebra and geometry — and are woefully underprepared for the modern world. The Los Angeles Unified School District is leading the way in updating the way math is taught. In 2013, the LAUSD secured approval from the University of California to recognize data science as a statistics course that students can substitute for Algebra 2 in the college pathway. Over 2,000 students are taking advantage of this option. The classroom we observed was full of critical thinkers who see data everywhere and appear comfortable interpreting, analyzing and questioning it. Modernizing math at a national level will require an intensive effort from educators, policymakers and high school counselors — as well as from students and parents who will need to advocate for it. Some states are already exploring changes to their mathematics frameworks, while a fair number of innovative teachers across the country are independently developing their own data-focused lesson plans. For this revolution to be carried out across the country, decision makers will need to hear from parents and other interested parties who recognize that our children deserve math instruction that is relevant to their lives.

Jo Boaler is a professor of mathematics education at Stanford University and author of “Limitless Mind.” Steven D. Levitt is a professor of economics at the University of Chicago and co-author of “Freakonomics.

Trigonometry Packet #15 page 2 of 6

From the Duke University Website 3.31.2020

Math Courses Required In Each Degree

• Biology

Lab Calculus 1 or Intro to Calculus 1

• Economics

Lab Calculus 1, Calculus II, Multivariate Calculus for Economics or Multivariate Calculus

• Math Degree

5 math courses above the basic calc 1, 2, 3 series and differential equations, PLUS Abstract Algebra, Advanced Calculus, Complex Algebra and One of many higher math courses totaling 15 courses.

• Environmental Science

Lab Calculus 1 or Lab Calculus 1

• Computer Science (calculus pre-requisite)

2 classes-Statistics, and one higher course of Calculus 3, Linear Algebra, Differential Equations or Matrices and Vectors.

• Physics

Calculus 1, Calculus 2, Linear Algebra, Differential Equation

• Engineering

6 course sequence in math and prob/stats including Calculus 1, 2, 3, Linear Algebra, Differential equations and one higher.

• English

1 (2 credit) quantitative analysis course

• Music

1 (2 credit) quantitative analysis course

• Theater Studies

1 (2 credit) quantitative analysis course

• Political Science 1 Statistics course

• History

1 (2 credit) quantitative analysis course

• Psychology

Quantitative Techniques (choice of courses), Calculus 2 or higher statistics course

Trigonometry Packet #15 page 3 of 6

The SAT Mathematics Section Breakdown of Questions

TOPICS

TOPICS

TOPICS

TOPICS

Trigonometry Packet #15 page 4 of 6

Use the “Opinion” Article Responses: ➢ Historical References:

The article mentions Sputnik, the Berlin Wall, the man on the moon, sock darning, and

shorthand. Choose 2 of these references and look them up on the internet or interview

someone who lived in the era they occurred. Document here 1) What was it and when did it

happen? 2) Why does the author use it in this article?

➢ Math Curriuclum Questions:

o Why do we use the algebra-geometry-algebra2-calc sequence in schools?

o What is the “proposal” that the author suggests and why is it obvious and

radical?

➢ Are you curious?

o What is Freakonomics? If you have internet, go check it out and tell me what you

find, here:

Trigonometry Packet #15 page 5 of 6

❖ Use the SAT TEST SUMMARY Reflections: o What stands out in the topics for the test?

o What topic categories are you taught the most in school? The least?

o How many questions are in the categories in which you feel most prepared? (as a percent)?

Questions in the categories where you feel the least prepared (as a percent)?

o If we stop teaching a strict algebra-geometry-algebra2-calculus curriculum and completely

teacher a data/statistics and real-life driven curriculum, will you be prepared for the SAT?

Why or why not? Is there research about that, and what does it say? There’s many; here’s

something: https://www.usatoday.com/story/news/education/2020/02/28/math-scores-high-school-

lessons-freakonomics-pisa-algebra-geometry/4835742002/ You may look at others to help you formulate broader thinking.

Use the Duke University Degree Programs Reflections:

What stands out to you about the courses required for different majors?

If we keep the algebra-geometry-algebra2-calculus track for all students, who will be

prepared for college? Why? Who may not be prepared for college? Why?

If we drop the algebra-geometry-algebra2-calculus track for all students, who will be

prepared for college? Why? Who may not be prepared for college? Why?

Are universities prepared to receive students with high school math courses that focus

around data/stats and not on the algebra-geometry-algebra2-calculus track? How do you

know?

Are we preparing ALL students for the math they need in college? Should we be? If not,

how can we?

Trigonometry Packet #15 page 6 of 6

Trigonometry Packet #16 page 1 of 1

Write a position essay of a minimum of 500 words on the topic of math curriculum in

schools. Some questions you should consider in your thinking are:

• Are you “under-prepared for the modern world”, as Jo Boaler suggests in her article?

• Is our math curriculum serving you well for your future plans? Is there something that you would like

to have learned to help you more for your future career?

• Is the algebra-geometry-algebra2-calculus track sufficient?

• Can there be a blending of curricula to include the algebra-geometry-algebra2-calculus track and a data

science/statistics track?

• What are things that hold schools back from changing what tracks they teach (based on our reading)?

• Do we need to “modernize math at the national level?” And if so, how?

• If we convert to a data/statistics-rich curriculum for all of our math courses, what are the pros, cons,

configurations of courses, and how should we do it?

Other considerations: • Make sure to write a full essay with proper form, grammar, paragraphs, complete sentences, and a

thesis/position made clear with points to back it up.

• You may use other resources to help you develop your thinking; make sure to cite them.

• Cite all sources you use, whether it is the 3 pieces we read here, or others. Cite websites if you use

them.

• Use your own thinking. There is no right or wrong answer or position. You don’t need to figure out

what I believe and write to it. Write your own opinion. I won’t grade you poorly if you disagree with

me. I won’t grade you higher if you agree with me. I will grade you based on your opinion and the

support you make for your points.

• Please don’t plagiarize. Cite the work you use. If you’re using someone else’s opinion, say so. You are

allowed to use viable authors to help you decide your opinion.

RUBRIC: 4 3 2 1 0

Position A strong position is evident in the essay.

A position is stated, but it is lacking conviction or consistency throughout.

A position is made but it doesn’t align with the prompts.

The essay discusses opinions on topic but the author’s opinion isn’t clear.

No position is evident in the essay.

Supporting Evidence

The essay has solid support with citations.

The essay has some support cited but not consistent.

The essay has statements of opinion in its structure but it isn’t supported by citations.

The essay has citations but they don’t align to the argument presented.

The essay has no support and no citations.

Format The essay is formatted well with well-placed paragraphs, grammar, complete sentences.

The essay is formatted well with well-placed paragraphs, some correct grammar, complete sentences.

The essay is somewhat formatted with well-placed paragraphs, some correct grammar and mostly complete sentences.

The essay is lacking a solid structure where points don’t align with paragraphs, sentence structure and grammar are lacking.

The essay has no solid format and poor grammar.