If you are curious what solid trigonometry is, this is for you. Often, CAD does not make this type of problem easier! Designers can save a lot of time if they can create the expression for the angles using pencil and paper. This is especially important if you are going to use it for a parametric relation later.
Trigonometric Function of General Angles LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Trigonometric Functions of Angles
Trigonometric Function Values
Could find the Six Trigonometric Functions
Learn the signs of functions in different Quadrants
Could easily determine the signs of each Trigonometric Functions
Solve problems involving Quadrantal Angles
Find Coterminal Angles
Learn to solve using reference angle
Solve problems involving Trigonometric Functions of Common Angles
Solve problems involving Trigonometric Functions of Uncommon Angles
If you are curious what solid trigonometry is, this is for you. Often, CAD does not make this type of problem easier! Designers can save a lot of time if they can create the expression for the angles using pencil and paper. This is especially important if you are going to use it for a parametric relation later.
Trigonometric Function of General Angles LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Trigonometric Functions of Angles
Trigonometric Function Values
Could find the Six Trigonometric Functions
Learn the signs of functions in different Quadrants
Could easily determine the signs of each Trigonometric Functions
Solve problems involving Quadrantal Angles
Find Coterminal Angles
Learn to solve using reference angle
Solve problems involving Trigonometric Functions of Common Angles
Solve problems involving Trigonometric Functions of Uncommon Angles
Derivation of a prime verification formula to prove the related open problemsChris De Corte
In this document, we will develop a new formula to calculate prime numbers and use it to discuss open problems like Goldbach, Polignac and Twin prime conjectures, perfect numbers, the existence of odd harmonic divisors, ...
Note: Some people found already errors in this document. I thank them for reporting them to me. Though, I am able to solve them, I deliberately want to keep these errors in the document for the time being to discourage error seekers from reading my papers. These people look at the details while missing the bigger picture.
Derivation of a prime verification formula to prove the related open problemsChris De Corte
In this document, we will develop a new formula to calculate prime numbers and use it to discuss open problems like Goldbach, Polignac and Twin prime conjectures, perfect numbers, the existence of odd harmonic divisors, ...
Note: Some people found already errors in this document. I thank them for reporting them to me. Though, I am able to solve them, I deliberately want to keep these errors in the document for the time being to discourage error seekers from reading my papers. These people look at the details while missing the bigger picture.
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Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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4. DO NOT MOVE ON UNTIL YOU HAVE ANSWERED THE QUESTION OR YOU NEED HELP!
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7. Graphing (Trigonometry) Ok, let’s start graphing this. The first thing you would have to choose is where the x-axis and the y-axis will go. In this graph, we’ll let the surface of the table be the x-axis (0 cm from the ground.) We’ll let the y-axis be the position of the net (0 cm from the net).
8. Graphing (Trigonometry) Next, we can plot the maximum and minimum points. The maximum y-value of this graph-to-be is 48 cm (info from question) and this happens when it is directly over the net. In our graph, the y-axis represents the position of the net. The minimum value of this graph is 0 cm (hits the table.). This happens 80cm away from the net in both directions.
9. Graphing (Trigonometry) Next we find the sinusoidal axis. That is just a fancy set of words that simply mean the average. The sinusoidal axis is not in a solid line because it is not actually part of the graph. To find it, the sinusoidal axis is the same as the midpoint line between the maximum and the minimum.
10. Graphing (Trigonometry) Now we can graph the points that are on the sinusoidal axis. These points are located halfway horizontally between the maximum points and the minimum points. In this case, the half distance between the maximum point and the minimum points is 40. This means that you would create points on the graph that are on the sinusoidal axis that are 40 units away from the x values of each maximum and minimum point.
11. Graphing (Trigonometry) Now we connect all the points we have created with a smooth curve. You graph may not look like this one but it should look close to this.
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17. Writing the Equation (Trigonometry) So let’s start with writing the sine equation. First we’ll find parameter D. Why not “A” you ask? D is just easier to find that’s all. So D is equal to the value of the sinusoidal axis. The equation of the sinusoidal axis in this case is y=24. This means the value of parameter D is 24.
18. Writing the Equation (Trigonometry) Next we’ll find parameter A. This value is the same as the amplitude. The amplitude is the distance between a max/min point and the sinusoidal axis. In this case, the distance between the sinusoidal axis and any one max/min point is 24. This means that the value of parameter A is 24. There is something else to know about A but we will talk about that after we find C.
19. Writing the Equation (Trigonometry) Next we’ll find parameter B. Parameter B is the same as the 2 π /(period of wave). The period of the wave is the length it takes for the wave to repeat. To find it, pick two points that are on the same part of the curve. This could be two consecutive mins, two consecutive maxes, or any two consecutive points that are on the same part of the graph. (ex. Any pair of same coloured points: black, purple, orange, red)
20. Writing the Equation (Trigonometry) To find parameter B, we’ll use the red dots (the min. values). The distance between the dots is 160. This means that the period of the wave is 160. This, however, does not mean that parameter B is 160. Parameter B is 2 π /(period of wave). Parameter B is equal to 2 π /160. Simplifying this fraction is optional. Keeping it has it’s benefits because from that, you can immediately see the period of the wave.
21. Writing the Equation (Trigonometry) Now for parameter C. Because the value of C is different for the sine equation and the cosine equation, we need to find them separately. First we’ll start with sine.
22. Writing the Equation (Trigonometry) We know that the graph of y=sin(x) starts at a “mid-point” in relation to the y-axis (It doesn’t start at the max value or the min value.) To find parameter C, we just look at how far any “mid-point” is from the y-axis. The value of C is different depending on what point you choose. In this example, we’ll use the first positive “mid-point”. This point is 40 units to the right of the y-axis so parameter C for the sine equation is 40.
23. Writing the Equation (Trigonometry) Now for cosine. We know that the graph of y=cos(x) starts at a maximum value. To find parameter C you look at how far any max value is from the y-axis. In this case we will use the closest max value to the y-axis. This value also happens to be on the y-axis so the value of parameter C is 0. You could also pick a minimum point. We will discuss that shortly.
24. The Issue with A (Sine) This isn’t really an issue but it could mess up your equation. When you choose your point that you are looking at, look at the way the graph passes through that point. The regular sine curve goes from the lower left to the upper right passing through this point. However, if the curve goes from the upper left to the lower right through your point, that means this graph has been flipped vertically and the amplitude of the wave is negative.
25. The Issue with A (Cosine) The issue with A being negative also appears in the cosine equation. The regular equation starts with a maximum point so if you pick a maximum point, your amplitude will be positive. However, you could also pick a minimum point to find the value of parameter C. However, picking a minimum point means that the graph of y=cos(x) has been flipped vertically. When you pick a minimum point to use as a reference, your amplitude will be negative.
26. Building the Equations Cosine A= 24 B= 2 π /160 C= 0 D= 24 Now we can start building the equations. Just take the basic form and input in all the parameters we just found. Sine A= -24 (explanation why A is negative: slide 24) B= 2 π /160 C= 40 D= 24 Your equations may be different depending on your parameters. However, the only parameter that should be different is parameter C.
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28. Solving the Equation (Trigonometry) This one is easy once you have the equation. Just plug and play! We’ll solve this equation using both the sine equation and the cosine equation we just built. She wants to hit the ball 106 cm from the net. We need to find the height at which the ball is when it is that far from the net. We let x=106 and we solve for y. Cosine Sine Mary has to have her paddle 11.46 cm from the surface of the table to hit the ball.
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30. Solving the Equation (Trigonometry) This one is like part c). However, now we are looking for the distance from the net, meaning we let f(x)=15 and we are looking for x. As you can see in the equation, the part in brackets looks pretty intimidating. For now, we can let θ =whatever is in the brackets and solve for θ . Cosine Sine
31. Solving the Equation (Trigonometry) Cosine Sine We are only using a decimal approximation but on your calculator you get a lot more decimal places. To keep this exact value, press [STO->][ALPHA][MATH] on your calculator to store that value as “A”. Use this exact value in your calculation for “x”. To use this value, press [ALPHA][MATH] on your calculator.
32. Solving the Equation (Trigonometry) Congratulations! You have found θ ! However, there are actually two answers for θ in each case. When sine is positive, the angle could be in quadrants 1 and 2. When cosine is negative, the angle could be in quadrants 2 and 3. To complete the answer, we need to find the related angles as well. The sine related angles are in green and the cosine related angles are in red. To find the other angle from the sine equation just take π - θ . To find the other angle from the cosine equation, take 2 π - θ . Great! We found all the θ values. Now we have to find “x”! Remember? We substituted θ for the part with the “x” earlier.
34. Solving the Equation (Trigonometry) Cosine So now we have two values of “x”. x= 49.7886, 110.2114
35. So how do we know which one to eliminate (if there is one to eliminate)?
36. Solving the Equation (Trigonometry) There is an answer we need to eliminate. Remember that this is a game of ping pong. You must let the ball hit the table first before you return. If we look at the graph, the ball hits the table at (80, 0). This means that the x value we are looking for is larger than 80. We can eliminate 49.7886 meaning that Mary must reach her arm out 110.2114 cm from the net to hit the winning shot!
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38. What are the Chances? There are two ways to do this question. One with a drawing and one using equations. It doesn’t matter which way you look at it. It’s like Mr.K’s block of wood. They are just different perspectives on the same question. First we’ll look at the method with drawing a tree. Well we know that there are three different types of shots Mary can make so let’s list them on the tree. Mary could also miss.
39. What are the Chances? We also know that after Mary makes her shot the Fiend will either return it or not return it. So let’s mark that down too!
40. What are the Chances? We also know some probabilities. Let’s mark it! P(Smash) = 1/15 P(Not returned if smashed) = 10/11 P(Lob) = 7/15 P(Returned if lobbed) = 7/11 P(Slice) = 6/15 P(Returned if sliced) = 3/11
41. What are the Chances? We can fill in the rest of the probabilities like so: P(Miss) = 1/15 (remaining part out of 15 from the 4 possibilities Mary has) P(returns if smashed)= 1/11 (because 10/11 chance of a miss) P(not returned if sliced) = 8/11 (because 3/11 chance of a hit) P(not returned if lobbed) = 4/11 (because 4/11 chance of a hit) P(returned if miss) = 0/11 (can’t return a ball that isn’t going towards you) P(not returned if miss) = 11/11 (can’t miss a ball that wasn’t returned to you)
42. What are the Chances? Now that we can calculate the probabilities of each scenario happening by multiplying the probabilities as we travel down each branch.
43. What are the Chances? To calculate the probability that Mary smashed the ball given the fact that the Fiend missed, the formula is: All we do is plug in the values and we find our probability!
44. What are the Chances? or P(smash|not returned)≈10.3093%
45. So the chances of Mary hitting the ball is approximately 10.3093%!
46. Click the word “Enjoy~!” At the beginning to move on to Episode 3!