1) What is this problem about?
This problem is about solving the trig of an inverse trig function and finding the value of the sum and difference formula without a calculator.
2) What must the reader pay close attention to in order not to make a mistake?
Readers need to pay close attention that to convert pi/6 to 30°, you have to multiply it by 180/pi and then reduce. Readers also need to pay attention what the coordinate values of a 30° angle is and how to find the missing side of a triangle when we only know two values. Readers also need to know what quadrant(s) the triangle is in because that will determine the number of answers you will receive in the end.
1) What is this problem about?
This problem is about using the sum and difference formulas when given values for a right triangle. We are given the values of the trig functions but we must be able to label it correctly on a triangle depending on which quadrant it is located in.
2) What must the reader pay close attention to in order not to make a mistake?
Reader must pay close attention to what quadrant the triangle is in. Sometimes we are given 2 trig functions and the answer in the end would contain two different answers but lucky for us, we know all three values and know what quadrant the triangle is in. Readers also must pay close attention to the type of signs used depending on what sum and difference formula we are using. Remember that you can only add and subtract with common denominators as seen in the numerator part of the tan difference formula.
1) What is this problem about?
This problem requires us to be able to find the exact values for the following trig functions and must learn to correctly choose an angle to works for all three equations
2)What must the reader pay close attention to in order not to make a mistake?
Readers must pay close attention to the fact that there are a large list of angles to choose from and they don't have to be restricted to using only one pair. Readers also must be careful with denominators when adding and subtracting because you can only add and subtract when there is a common denominator.
Step1 : set cot(x)= to its ratio which is x/y.
Step2: Find the reference angle that makes the equation equal to 1. So the only one that works is 45°.
Step3:Determine the quadrants we will have our answers in. Since cot(x)= positive answer the x and y values must both be either positive or negative. So the quadrants that work for that is 1st and 3rd.
Step4:To find the angle in quadrant1, we just use the RA which was 45° and to find the angle in the third, take the RA and add it to 180° which makes it 225°.
Step5:Convert our degree answers to radians by multiplying 45° and 225° by pi/180°.
Step1:Isolate the csc(x) by dividing both sides by √3 and then rationalize toget csc(x)=2√3/3.
Step2 : set cot(x)= to its ratio which is x/y.
Step3: Find the reference angle that makes the equation equal to 2√3/3. So the only one that works is 60°. Since r=1 and dividing by a fraction is the same thing as multiplying by its recip.
Step4:Determine the quadrants we will have our answers in. Since csc(x)= positive answer, the y values must positive. So the quadrants that work for that is 1st and 2nd.
Step5:To find the angle in quadrant1, we just use the RA which was 60° and to find the angle in the second, take the RA and subtract it from 180° which makes it 120°.
Step6:Convert our degree answers to radians by multiplying 60° and 180° by pi/180°.
Step1:The problem we are looking at looks similar to a quadratic equation so we can substitute sin(x) for u. After factoring it out, plug back in the sin(x) for all u-variables.
Step2 : Now we want to use Zero Product Property and set sin(x)= to its ratio which is y/r and make it equal to 0.
Step3: After solving for the ratio values, we see that y=0 and -1.
Step4:Determine the quadrants we will have our answers in. Since y=0 and -1, then we are looking at quadrant angles and those angles are 0°,360°,180°, and 270°.
Step5:Convert our degree answers to radians by multiplying 0°,360°,180°, and 270° by pi/180°.
Step1:We don't want to work with the 2x attached to the cos yet so we are going to sub it out for the variable(u).
Step2 : set cos(x)= to its ratio which is x/r.
Step3: Find the reference angle that makes the equation equal to -√3/2. So the only one that works is 30°.
Step4:Determine the quadrants we will have our answers in. Since cos(u)= negative answer, then the x-value must be a negative. So the quadrants that work for that is 2nd and 3rd.
Step5:To find the angle in quadrant2, we just take the RA and subtract it from 180° which makes it 150°. and to find the angle in the third, take the RA and add it to 180° which makes it 210°.
Step6:Since u=150° and 210°, plug back in the 2x for the u variable and solve for the unknown variable, so our final answers should be 75° and 105°.
To begin with concept 2, we are given a problem(right) that has tan(x)=7/24 and sec(x)=-25/24. The ratio for tan is y/x and since it equals a positive answer, then the quadrants we should now be look at is 1st and 3rd since +y/+x=+ answer and -y/-x=+answer. The ratio for sec is r/x and since r is always equal to 1 and the ratio equaled a negative value, then the value of x must be negative so we would be looking at 2nd and 3rd quadrant. The only quadrant both of these trig functions have in common is the third, so the answers should be located in the area of the 3rd quadrant. Now to solve for the other trig functions. Cot(x)=1/tan(x). Plug in what we know so cot(x)=24/7. Cos(x)=1/sec(x), so plug in the sec(x) value and we get cos(x)=-24/25. Now we will use a ratio identity in which tan(x)=sin(x)/cos(x). Multiply the cos(x) over to the tan(x) side to solve for sin(x). Plug in the values we now know and we get that sin(x)=-7/25. Csc(x)=1/sin(x), so plug in the new sin(x) value that we just found and we see that csc(x)=-25/7.
Now to solve using triangles(left). We already know that answers should be in the third quadrant is the x and y-values should be negative. Tan(x) ratio is y/x and that is equal 7/24 therefore we can deduce that y=-7 and x=-24 since the 3rd quadrant forces the y and x values to be negative. Sex(x) is r/x which is equal to -25/24. Since we know that x=-24, we just got rid of the denominator and the negative sign therefore leaving behind r=25. Using the ratios of all the other trig functions, we can find their values.
The very first Pythagorean theorem is sin^2(x)+cos^2(x)=1 and the reason is because if we write out all the variables we see that [(y^2)+(x^2)]/r^2=1. Multiply the r^2 over to the other side so its [(y^2)+(x^2)]=r^2. r is always equal to one on a unit circle therefore the first identity is sin^2(x)+cos^2(x)=1.
To derive the second identity, we first divide everything by sin^2. Sin^2 divided by itself is always equal to 1 and now we use ratio and reciprocal identities for the rest. cos^2/sin^2 is equal to cot^2 due to ratio identity and 1/sin^2=csc^2 because of reciprocal identity.
To derive the third identity, we do the exact same thing except we now divide everything by cos^2. Cos^2 divided by itself is always equal to 1 and now we use ratio and reciprocal identities for the rest. sin^2/cos^2 is equal to tan^2 due to ratio identity and 1/cos^2=sec^2 because of reciprocal identity.
1. How have you performed on the Unit O and P tests? What evidence do you have from your work in the unit that supports your test grade (good or bad)? Be specific and include a minimum of three pieces of evidence.
I've been preforming quite well and the evidence of that is I've been scoring A's on both of them. Even though you stopped checking WSQ charts, i still continue to complete as least 3/4 of the SSS packet to make sure i know how to do the concepts. PQs give me a self assessment if i do understand and Quizzes are what determine if i still remember how to do that certain concept when the test comes by.
2. You are able to learn material in a variety of ways in Math Analysis. It generally follows this pattern:
→ Your initial source of information is generally the video lessons and SSS packets followed by a processing and reflection activity via the WSQ
→ individual supplemental research online or in the textbook before class
→ reviewing and accessing supplementary resources provided by Mrs. Kirch on the blog
→ discussion with classmates about key concepts
→ practice of math concepts through PQs
→ formatively assessing your progress through concept quizzes
→ cumulatively reviewing material through PTs
→ Final Assessment via Unit Test.
Talk through each of the steps given in the following terms:
a. How seriously do you take this step for your learning? What evidence do you have to support your claim? Make sure to make reference to all 8 steps.
b. How could you improve your focus and attention on this step to improve your mastery of the material? What specific next steps would this entail? Make sure to make reference to all 8 steps.
I take a keen liking to math and understanding of these units because my future as an engineer will require a lot of math and science.
1)I like to learn through self trial and error before i do any videos. By doing that, it helps set my mind to the college life because out there, you are on your own most of the time. If i don't understand or am clueless, i'll alter all of my attention to the youtube videos and explanations of the unit instead of fussing and boiling up inside.
2)Watching youtube videos of Mrs.Kirch has been my individual supplemental research online and i usually don't go searching for other inputs or instructions because usually I understand how to do a certain concept after one video. But occassionally, when i still don't understand from Mrs. Kirch's vids, i would go to other youtube videos and instructors and observe their deductive reasoning in solving a certain set of problems.
3)Linking my phone to edmodo has been one of the most useful but sometimes annoying thing to happen this year. I am able to receive notices that reminds me to check out some of the things Mrs. Kirch posted because usually anything Mrs. Kirch posts or talks about is probably important.
4)WSQ chats in class have been very useful because it helps refreshes the mind of what it learned the previous day and if a student doesn't understand, then another student can teach. Personally to me, teaching and tutoring somebody else is the best way of memorization. But sometimes i don't understand a certain concept and alter my focus in participating in the small groups done in class. By doing that, i am able to get a visual, auditory, and sensual explanation of the concept and as a sensual and visual learner, i will be able to mastery that material.
5)PQs has always been my way of checking my methods and making sure that I understand how to do the concept. Getting the right answer is one thing, but i prefer being wrong most of the times. Through that, i am able to observe my mistakes and grab a better understanding of where i went wrong the understand why i must go down this path instead of another.
6)As stated above in the first paragraph, quizzes are there to make sure that i understand and memorize how to do certain type of problems. By getting 8s, i become more confident for the upcoming test but i do receive anything lower than an 8, i must change my focus on redo-ing SSS problems or PQs or attend 1/group-on-1 tutoring with Mrs. Kirch till i feel confident that i will get an 8 on the quizzes.
7)PTs has always been that review needed to help refresh the mind and make sure that these concepts will remain with you for a part of your high/college life.
8)The day has come. Unit tests is the ultimate judging of how well i did in my studying and gauges how much effort i have put into my education. My goal is always to get a 100, but if i receive an A, thats good for me but there is always much more to learn. If i get a low test score, i shall attend after-school or lunch tutorings to make sure i do better on the retake(which i will never do, cause i'll make sure i will never have to retake any tests. HAHAHA)
3. Reflect on your learning this year thus far by considering the following questions:
a. How confident do you generally feel on the day of a Unit Test? Give evidence and specifics to back up your answer.
b. How well do you feel you have learned the math material this year as compared to your previous years in math? Give evidence to support your claim.
c. How DEEPLY do you feel you have learned the math material this year as compared to your previous years in math? Give evidence to support your claim.
d. Do you normally feel like you understand the WHY behind the math and not just the WHAT/HOW? Meaning, do you understand why things work, how they are connected to each other, etc, and not just the procedures? Explain your answer in detail and cite specific evidence from this year.
e. How does your work ethic relate to your performance and success? What is the value of work ethic in real life?
I dont gauge my confidence till i see the very first problem. If i know how to solve it and get the answer, i will be confident throughout the test on that day. Confidence=level of understanding to me and if i understand most of the test, my confidence level is way up there and that pays off with the A's i make sure to receive on each unit test. This new system of flipped classroom has caught me off guard cause i've never experienced such a thing in school. But i feel like i learn much more from this system because i can spend as long as i want at home and the only thing being discuss in class will be questions. I learned A LOT from this teaching system and i seem to benefit a lot from it because of the grades i've been receiving. All that hard work is slowly being paid off. im the type of person who doesn't care about the answer but the process of how to get there and why this process is chosen or why it works. Answer is easy to copy and get from other people, but understanding why and how to get to that answer will greatly aid in one's learning in the future. My work ethic is somewhat constant and usually on the day before the unit test, i don't even touch math because i live by "either you know it or you don't. last minute cramming is not going to get you anywhere and you will slowly regress all that information in less than a week or so."
