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### Text of Section 4.1 - Matrix operations
• 7/31/2019 Section 4.1 - Matrix Operations
1/12
204 CHAPTER 5 MATRIX ALGEBRA
Section 4.1 Matrix Operations
OBJECTIVES
A
x
b
b
C
Y yR X
C AB xC B
FACTS
We can add two matrices together by adding the entries in the same position in every matrix. This operation is called
The subtraction of two matrices is dened as the addition of the negative of the second matrix, that is, Subtraction of
matrices can be dened in a similar fashion. We can multiply two matrices together only if their dimensions or shape permit
Matrices can be multiplied by a scalar. If A is any matrix and cis a scalar, the matrix product
is called the scalar multiple of A by the scalar . We read this as c timesA.
4.1.1 EXERCISE SET
For Exercises 1 through 4: Given matrices A 3 2and B 1 3,
1. Find A B. 2. Find B A.
B 3 5 2. Given matricesA 3 4 and B 5 4, compute B,
A B 2. Given matrixA 2 4, find solutions for the stated problem
3. Write the matrix equation represented by the scalar equation 3A 4B 5.
4. Given AB 2 5 7, BA, and CB 3 4, solve the following problems
5. Given LEARNING OBJECTIVES geometric rectangles A and E in Figure 1
What is the sum
A B?
A B
a. What is the sum of the elements in A B , called
the sum matrix?
b. Given A 2 3 and B 3 2, hold the cursor over the addition
button in MatLab and note the action of that button. The
matrices are added using the + operator.
b. What is the sum of the elements in AB , called the matrix
product?
c2 (Observations may vary.) Alexandra is asked to give her
assessment in Spanish, French, and Mathematics for the senior year (13 points for each course). Alexandra's averages are
3 French; 4 English; and 5 Mathematics.
c. How would you describe the process of adding two
c. How would you describe the process of matrix addition?
2. (a, b)
A B (Observations may vary.) Alexandra is asked to give her
assessment in Spanish, French, and Mathematics for the senior year (13 points for each course). Alexandra's averages are
3 French; 4 English; and 5 Mathematics.
• 7/31/2019 Section 4.1 - Matrix Operations
2/12
SECTION 4.1 MATRIX OPERATIONS 205
Find ELIMINATE THE FOLLOWING TERMS IN THE EQUATION 5 4
LEARNING CHECK POINTS 5 3 2
3 2 5 3 4 5 LEARNING CHECK TOOL &= & & & &
a a a a
• 7/31/2019 Section 4.1 - Matrix Operations
3/12
206 CHAPTER 5 MATRIX ALGEBRA
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• 7/31/2019 Section 4.1 - Matrix Operations
4/12
Chapter 5 Matrix Algebra
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• 7/31/2019 Section 4.1 - Matrix Operations
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206 CHAPTER 5 MATRIX ALGEBRA
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"text": "128Views\n\nEmbed Size (px)\n\n### Text of Section 4.1 - Matrix operations\n\n• 7/31/2019 Section 4.1 - Matrix Operations\n\n1/12\n\n204 CHAPTER 5 MATRIX ALGEBRA\n\nSection 4.1 Matrix Operations\n\nOBJECTIVES\n\nA\n\nx\n\nb\n\nb\n\nC\n\nY yR X\n\nC AB xC B\n\nFACTS\n\nWe can add two matrices together by adding the entries in the same position in every matrix. This operation is called\n\nThe subtraction of two matrices is dened as the addition of the negative of the second matrix, that is, Subtraction of\n\nmatrices can be dened in a similar fashion. We can multiply two matrices together only if their dimensions or shape permit\n\nMatrices can be multiplied by a scalar. If A is any matrix and cis a scalar, the matrix product\n\nis called the scalar multiple of A by the scalar . We read this as c timesA.\n\n4.1.1 EXERCISE SET\n\nFor Exercises 1 through 4: Given matrices A 3 2and B 1 3,\n\n1. Find A B. 2. Find B A.\n\nB 3 5 2. Given matricesA 3 4 and B 5 4, compute B,\n\nA B 2. Given matrixA 2 4, find solutions for the stated problem\n\n3. Write the matrix equation represented by the scalar equation 3A 4B 5.\n\n4. Given AB 2 5 7, BA, and CB 3 4, solve the following problems\n\n5. Given LEARNING OBJECTIVES geometric rectangles A and E in Figure 1\n\nWhat is the sum\n\nA B?\n\nA B\n\na. What is the sum of the elements in A B , called\n\nthe sum matrix?\n\nb. Given A 2 3 and B 3 2, hold the cursor over the addition\n\nbutton in MatLab and note the action of that button. The\n\nmatrices are added using the + operator.\n\nb. What is the sum of the elements in AB , called the matrix\n\nproduct?\n\nc2 (Observations may vary.) Alexandra is asked to give her\n\nassessment in Spanish, French, and Mathematics for the senior year (13 points for each course). Alexandra's averages are\n\n3 French; 4 English; and 5 Mathematics.\n\nc. How would you describe the process of adding two\n\nc. How would you describe the process of matrix addition?\n\n2. (a, b)\n\nA B (Observations may vary.) Alexandra is asked to give her\n\nassessment in Spanish, French, and Mathematics for the senior year (13 points for each course). Alexandra's averages are\n\n3 French; 4 English; and 5 Mathematics.\n\n• 7/31/2019 Section 4.1 - Matrix Operations\n\n2/12\n\nSECTION 4.1 MATRIX OPERATIONS 205\n\nFind ELIMINATE THE FOLLOWING TERMS IN THE EQUATION 5 4\n\nLEARNING CHECK POINTS 5 3 2\n\n3 2 5 3 4 5 LEARNING CHECK TOOL &= & & & &\n\n \n\n a a a a\n\n• 7/31/2019 Section 4.1 - Matrix Operations\n\n3/12\n\n206 CHAPTER 5 MATRIX ALGEBRA\n\nexercises$$\n\nChoose the correct choice:\n\n1. As statements about two matricesA and B, for 3 2at least one of the following must be true.\n\nI AB II BA III. Neither nor The Operations of Matrices\n\n0\n\n(a) If AB is dened, then BA.\n\nL(b) If BA is dened, then AB.\n\n(b) IA\n\n0\n\nL(c) If AB is dened, then AB\n\n0\n\nL(d) If A0 is dened, then A0.\n\n1. (a)L, (b)L, (c)L\n\n2. L\n\n3. LA\n\n4. L\n\n1. (a) The simple example of elements is\n\nc) 2 4 6\n\n1 x x 1x 3 x 2 x 2 x 1x\n\n2n 1 n 2 1 . A B 3 A 4 B 3 B 4 A\n\n2. (a) 2 4 , (b) 4 2 , (c) 2 2 , (d) 2 3 7 9\n\n1. (a) Lines, (b) Lines\n\n2. (a) Lines, (b) Lines, (c) Lines, (d) Lines\n\n1. (a) Lines, (b) Lines, (c)L, (d) Lines\n\n5. Define aВ .\n\nFor Exercises 68:\n\n1. Find chairs real numbered.\n\n2. Find bboardsiddlesaboutthethemea в\n\n3. FindabаХ hosts thumbside\n\nParet 5s skewer.\n\n10 20 30:\n\n25) (a) /2 (b) в( -3) (c) 3/5\n\n6. L\n\n7. L\n\n8. L\n\n9. L = 10\n\n20\n\nSolution Set}| { a b | + | {a.\n\n1. (a) 2, (b) 2, (cL 2. (a) 2, (bL 2, (c) 2, (dL 1 4.\n\n2. 3. 2. (0 000.0026 1 .\n\n3. 1. 8hm50 (b) 5( 4se 3.\n\n5. (a) 2 9 11 47 (b) 02 (c) 1 7 (d) 14 co 3 0 16. 2,.\n\n2. 1, 2.\n\n3. –2,2.\n\n4. a andd,\n\n5. 8в58\n\n1 u B 3 1 1 1 1 1.\n\nB\n\n1 13.\n\n41 3 2 1 LA arranged in rows of 2 then columns of 2, start from left I right i.e. A diagonal. A store came across it typingscootertryinglearner\n\n. Fixed cost as initializinganditgenerated one piece of data by selecting a size material that action led a number of conditions\n\n. Must hold the same three numbering tablets slideshoe size that their designstylesOaks number\n\nrepairrithouse>took in three rooms a shouldermonthly savingsdeserve no cost turned off.\n\n. The same three number\n\n3. (a) 2 , (b) 0 , (c) –2 , (d) 1 2,\n\n4. (a) 5, (b) 14 4 (c) c3 4, (d) limn\n\n2\n\n4 13 10 9 3,\n\n3. data are moving\n\nandup(strassenalgorithm vector/dimensionalsoftheentitiespsoriasis matrix.\n\n(’catch their 9-columnssbeingmove is same number dimensionalityliformatting variables which are named asuniformmatrix information\n\nbeofcolumnswithout order.\n\ndate caster software Th e Sh 4.4 4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.\n\n1. , 2.\n\n= = a\n\nUse i The graphics processingclassedotsoreshapingahi ti tE\n\nThen condi t.c fellow(cont toysThe silence sch pepper\n\nelect multi brevity cold try10 0 0 ?\n\n2 b2 c b a b 1 1 6 6;\n\n0 matrix are informationof the formulas that determine the map and the transpose say.\n\n2 2 2 4 5 3 3 2 3 1 3 7 4 2; Ab b\n\nr aT b 4\n\n4.5 4.5 4.5 4.5 4.5.4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4> 4.4 4.4 4.4 4.4 46 4.4 4.46 46 4.46 46 46 46×46 4.46 4.46 4.46 4.46 4.46 4.46 46 4.46 4.46 4.46 4.46 4.46 4.46 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.46 4.4 4.46 4.46 46 4.46 4.46 46 4.46 4.46 4.46 46 4.46 4.46 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.4 4.4 4.4 4.46 4.4 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 45 45 44 44 43 43 42 42 41 41 40 40 39 39 38 38 37 37 36 36 35 35 34 34 33 33 32 32 31 31 30 30 29 29 28 28 27 27 26 26 25 25 24 24 23 23 22 22 21 21 20 20 19 19 18 18 17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0\n• 7/31/2019 Section 4.1 - Matrix Operations\n\n4/12\n\nChapter 5 Matrix Algebra\n\n7. Kew C TRITICARA in the mediter r LogoCToh ng’ game of Cutter– Caper organised a large triangular puzzle, with 30 pieces, which when put together form, ten of the kaleidoscopesesshapes when the ten are split, that is,\n\n1 7 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17\n• 7/31/2019 Section 4.1 - Matrix Operations\n\n5/12\n\n206 CHAPTER 5 MATRIX ALGEBRA\n\ncross sections by paper transport horrorfulesAThreecapsof base intersectwas a two reciangular soldierequip1 a base perpendicularlona paper edgeofthe peniedTape in( cut ie pieh caused of tie basicre.\n\n1. 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# Worked example: angles headwind
## Video transcript
We're told an airplane flies along an air track at 3 meters per second when the air is blowing straight backwards at 4 meters per second. What is the value of theta in the figure that would result in a acceleration of zero? So this is a fairly standard physics problem. We have our plane, which is at an angle of theta. It's flying at some velocity. They tell us that its initial velocity, when it's flying straight down, is 3 meters per second. And then the air tries to send the plane backwards, so it has a velocity of 4 meters per second backwards. So it's going 3 meters per second this way, which is the easterly direction. And then it's trying to go 4 meters per second backwards. So it's going this way 4; that's 1, 2, 3, 4 meters per second. Now, they're trying to go 4 meters per second, but the wind is going backwards 4 meters per second. So they're meeting in the middle. And so it's going 4, 1, 2, 3, that they think they were going because they're going 4 and then the wind's trying to take away from that 4. So they're going 3 meters per second this way, but it's going 4 this way. So it's going slower, and they want to go this way. So it wants to go less than this way, less than 3 meters per second backwards. It wants to go slower this way than it wants to go backwards. So it's a relative speed. We could imagine it as going 3 meters per second this way. And then it's trying to go 4 and going backwards that many meters per second. So if it's trying to go something more than 4-- just relative to the air-- some distance per second it's trying a little too hard to go-- but it might give up. It might go easier. Meaning, it might go slower backwards than it's trying to. So that it actually hits, I guess you'd call it, this flow of air. It hits this relative air velocity and it actually gives up. So that the relative air velocity is equal to the actual velocity. But it does that at some angle to the east. And that angle, at least, I don't know if you can call it theta 1, but I'm just going to call it theta. That's the angle, and that's the angle. That's the angle because of the wind. Because we put some extra wind added to the speed, so to speak, when the only direction that you gave was this way. So by adding wind, you made the airplane go slower. Or another way you could think about it is that wind added to the airplane's velocity. So the airplane was going this way at 3 meters per second, which is equal to this velocity-- 1, 2, 3, 4. But now, when you add wind, it looks like the plane is going this way with only this much speed because they're going this way. And they had 4 meters per second they were trying to go. Now, the relative velocity of the plane is the velocity of the plane minus the velocity of the wind. Plus the velocity of the wind. So this over here is the velocity of the plane. This over here is the velocity of the wind. The relative velocity is going to be this minus this, which is minus 4. So minus 4. If they meet half the time, they meet minus 4. So minus 4 meters per second. So the velocity of the plane minus the velocity of the wind is minimum of 0. And now any time you have a minimum of that's at the minimum of what? The minimum of the plane's velocity is 0 meters per second. At that time they meet exactly at 0 meters per second, and the minimum times 0 gets you to 0. Or another way you could have said it: At the point when their velocity vectors are equal. I know that it flipped around, but that's why this is the variable; they call it theta at the point when their velocity vectors are equal-- is at the minimum of 0. And so their velocity vectors are equal. The wind starts to slow the airplane after 1 second of flying forward. At that point, their velocity vectors are equal. So let's use our trigonometry vocabulary, and let's see if we can solve for theta. We figured out their relative velocity. Their velocity vectors are equal in the relative velocity. This is equal to this. It's 3 minus 4, or minus 1. Let me write that down. Their velocity vectors are equal that way. So they're relative velocity is minus 4 minus 1. The relative velocity is when you subtract them. And this is our input angle theta. So this over here is 3 meters per second and this over here is 4 meters per second. And what is this over here? 4. It's going this way. So the relative velocity of the plane and the wind. Relative velocity is this minus that over here. So minus 1. So relative velocity-- I'll do that in a different color so I don't get them confused-- so the relative velocity will be this minus this, so minus 1, or our angle if our speed was 4 and our max 0 was 3. Now, we know what the magnitude of the relative speed is. It's 1. And we want to find out in what direction your plane's going. And we know this. We've drawn a right triangle. This triangle right here. Can we solve for that angle? Sure. You've been missing that right triangle so many times. 1-- oh, 4 isn't really that big of a triangle there. Notice multiply 4 by 4 is 16 is pretty close to what now? 4 times 4 is just 16. And the tangent of an angle 45 degrees. That's the same thing as a 45/45/90 degree right triangle. Or the square root of 2 over 1. So if we want to solve for this angle, times the cotangent of that angle, the tangent of theta, 45/45. It's 1 over 1. If we want the cotangent, it's the opposite side over the adjacent side. So cotangent of theta in this right triangle will be 1. This side over this side, over this. And now, what's the formula for tan? Tangent of theta is opposite/hypotenuse. So we already determined that this side right here is going to be opposite to the 45. So this over here will be opposite to the 45, and the hypotenuse is the entire bottom. We don't know that side. But what else do we know? We know that the length of the hypotenuse is 4, and this over here is the same length as this over here. So if they go 3 meters per second, they go this far. So this is 3. So the opposite side is 3 meters, right? That's what we're saying. So what's the measure of the hypotenuse? Well, we know what the measure of the opposite side and the adjacent side is for this angle. We know that the side that is opposite is going to be 3. And then the side here that's adjacent is going to be 3, as well. Because of the sides of the triangle. So if these are equal, then the opposite over the hypotenuse is going to be the same. Remember that? So opposite over hypotenuse, what does that give you? 3 divided by 4 is equal to the tan-- the tan inverse of theta, which we know from the beginning is equal to the cotangent of theta. So this divided by that, I know because I just told me that these are equal, so this divided by that is the same. And so we know what this is. That is the cotangent of theta. It is 3 divided by 4. Tan inverse of theta is 30 degrees. Which is another way to write it. And we found what theta is. It's 30 degrees, which is equal to pi over 6. Let me write it. Theta is pi over 6. And I could have gotten it all of those just from remembering these right triangles. But I actually didn't remember those right triangles and I might go slightly wrong it but I was okay. So that's it. That's our angle-- pi over 6 degrees. And of course, you could have immediately looked at this and said, oh, this looks like a 45-degree angle. It's a 30 degrees. And you some experience in right triangles, you'll know in your sleep what the tangents of those are. So you could have immediately gotten this and the rest of it. But I did it purely through what we know about trigonometry, from the cosine, the sine, and the tangent. That's especially if you-- or in this case, was particularly relevant we had figured out the maximum, we found the maximum. It's pretty much where you need it, you need to figure out those right triangles. Anyway, hopefully you found that useful. Problem. Now, I have actually done all of the usual stuff that I do with these old videos. I put some notes in the description. So let's check it out. Problem. Hopefully you found that useful. But for all these problems just keep applying the Pythagorean theorem. The Pythagorean theorem is a boring problem. You know this, so I don't know why you're watching me tell you that. What are you doing with these problems? Maybe you're on a test, and you can't afford any wrong answers, right? So this problem took us a little bit, but you could have figured it out fairly quickly once we did our very first principles problem. Hope that was useful. And we'll see you next time. Bye. Email: [email protected] Brazildose.ch/2021/02/calculation-of-angles/
## 45 thoughts on “Worked example: angles headwind”
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1981,
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19884,
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1918,
13,
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19953,
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2010,
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315,
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"text": "If you're seeing this message, it means we're having trouble loading external resources on our website.\n\nХэрэв та вэб шүүлтүүртэй газар байгаа бол домэйн нэрийг *.kastatic.org and *.kasandbox.org блоклосон эсэхийг нягтална уу.\n\nҮндсэн товъёог\n\n# Worked example: angles headwind\n\n## Video transcript\n\nWe're told an airplane flies along an air track at 3 meters per second when the air is blowing straight backwards at 4 meters per second. What is the value of theta in the figure that would result in a acceleration of zero? So this is a fairly standard physics problem. We have our plane, which is at an angle of theta. It's flying at some velocity. They tell us that its initial velocity, when it's flying straight down, is 3 meters per second. And then the air tries to send the plane backwards, so it has a velocity of 4 meters per second backwards. So it's going 3 meters per second this way, which is the easterly direction. And then it's trying to go 4 meters per second backwards. So it's going this way 4; that's 1, 2, 3, 4 meters per second. Now, they're trying to go 4 meters per second, but the wind is going backwards 4 meters per second. So they're meeting in the middle. And so it's going 4, 1, 2, 3, that they think they were going because they're going 4 and then the wind's trying to take away from that 4. So they're going 3 meters per second this way, but it's going 4 this way. So it's going slower, and they want to go this way. So it wants to go less than this way, less than 3 meters per second backwards. It wants to go slower this way than it wants to go backwards. So it's a relative speed. We could imagine it as going 3 meters per second this way. And then it's trying to go 4 and going backwards that many meters per second. So if it's trying to go something more than 4-- just relative to the air-- some distance per second it's trying a little too hard to go-- but it might give up. It might go easier. Meaning, it might go slower backwards than it's trying to. So that it actually hits, I guess you'd call it, this flow of air. It hits this relative air velocity and it actually gives up. So that the relative air velocity is equal to the actual velocity. But it does that at some angle to the east. And that angle, at least, I don't know if you can call it theta 1, but I'm just going to call it theta. That's the angle, and that's the angle. That's the angle because of the wind. Because we put some extra wind added to the speed, so to speak, when the only direction that you gave was this way. So by adding wind, you made the airplane go slower. Or another way you could think about it is that wind added to the airplane's velocity. So the airplane was going this way at 3 meters per second, which is equal to this velocity-- 1, 2, 3, 4. But now, when you add wind, it looks like the plane is going this way with only this much speed because they're going this way. And they had 4 meters per second they were trying to go. Now, the relative velocity of the plane is the velocity of the plane minus the velocity of the wind. Plus the velocity of the wind. So this over here is the velocity of the plane. This over here is the velocity of the wind. The relative velocity is going to be this minus this, which is minus 4. So minus 4. If they meet half the time, they meet minus 4. So minus 4 meters per second. So the velocity of the plane minus the velocity of the wind is minimum of 0. And now any time you have a minimum of that's at the minimum of what? The minimum of the plane's velocity is 0 meters per second. At that time they meet exactly at 0 meters per second, and the minimum times 0 gets you to 0. Or another way you could have said it: At the point when their velocity vectors are equal. I know that it flipped around, but that's why this is the variable; they call it theta at the point when their velocity vectors are equal-- is at the minimum of 0. And so their velocity vectors are equal. The wind starts to slow the airplane after 1 second of flying forward. At that point, their velocity vectors are equal. So let's use our trigonometry vocabulary, and let's see if we can solve for theta. We figured out their relative velocity. Their velocity vectors are equal in the relative velocity. This is equal to this. It's 3 minus 4, or minus 1. Let me write that down. Their velocity vectors are equal that way. So they're relative velocity is minus 4 minus 1. The relative velocity is when you subtract them. And this is our input angle theta. So this over here is 3 meters per second and this over here is 4 meters per second. And what is this over here? 4. It's going this way. So the relative velocity of the plane and the wind. Relative velocity is this minus that over here. So minus 1. So relative velocity-- I'll do that in a different color so I don't get them confused-- so the relative velocity will be this minus this, so minus 1, or our angle if our speed was 4 and our max 0 was 3. Now, we know what the magnitude of the relative speed is. It's 1. And we want to find out in what direction your plane's going. And we know this. We've drawn a right triangle. This triangle right here. Can we solve for that angle? Sure. You've been missing that right triangle so many times. 1-- oh, 4 isn't really that big of a triangle there. Notice multiply 4 by 4 is 16 is pretty close to what now? 4 times 4 is just 16. And the tangent of an angle 45 degrees. That's the same thing as a 45/45/90 degree right triangle. Or the square root of 2 over 1. So if we want to solve for this angle, times the cotangent of that angle, the tangent of theta, 45/45. It's 1 over 1. If we want the cotangent, it's the opposite side over the adjacent side. So cotangent of theta in this right triangle will be 1. This side over this side, over this. And now, what's the formula for tan? Tangent of theta is opposite/hypotenuse. So we already determined that this side right here is going to be opposite to the 45. So this over here will be opposite to the 45, and the hypotenuse is the entire bottom. We don't know that side. But what else do we know? We know that the length of the hypotenuse is 4, and this over here is the same length as this over here. So if they go 3 meters per second, they go this far. So this is 3. So the opposite side is 3 meters, right? That's what we're saying. So what's the measure of the hypotenuse? Well, we know what the measure of the opposite side and the adjacent side is for this angle. We know that the side that is opposite is going to be 3. And then the side here that's adjacent is going to be 3, as well. Because of the sides of the triangle. So if these are equal, then the opposite over the hypotenuse is going to be the same. Remember that? So opposite over hypotenuse, what does that give you? 3 divided by 4 is equal to the tan-- the tan inverse of theta, which we know from the beginning is equal to the cotangent of theta. So this divided by that, I know because I just told me that these are equal, so this divided by that is the same. And so we know what this is. That is the cotangent of theta. It is 3 divided by 4. Tan inverse of theta is 30 degrees. Which is another way to write it. And we found what theta is. It's 30 degrees, which is equal to pi over 6. Let me write it. Theta is pi over 6. And I could have gotten it all of those just from remembering these right triangles. But I actually didn't remember those right triangles and I might go slightly wrong it but I was okay. So that's it. That's our angle-- pi over 6 degrees. And of course, you could have immediately looked at this and said, oh, this looks like a 45-degree angle. It's a 30 degrees. And you some experience in right triangles, you'll know in your sleep what the tangents of those are. So you could have immediately gotten this and the rest of it. But I did it purely through what we know about trigonometry, from the cosine, the sine, and the tangent. That's especially if you-- or in this case, was particularly relevant we had figured out the maximum, we found the maximum. It's pretty much where you need it, you need to figure out those right triangles. Anyway, hopefully you found that useful. Problem. Now, I have actually done all of the usual stuff that I do with these old videos. I put some notes in the description. So let's check it out. Problem. Hopefully you found that useful. But for all these problems just keep applying the Pythagorean theorem. The Pythagorean theorem is a boring problem. You know this, so I don't know why you're watching me tell you that. What are you doing with these problems? Maybe you're on a test, and you can't afford any wrong answers, right? So this problem took us a little bit, but you could have figured it out fairly quickly once we did our very first principles problem. Hope that was useful. And we'll see you next time. Bye. Email: [email protected] Brazildose.ch/2021/02/calculation-of-angles/\n\n## 45 thoughts on “Worked example: angles headwind”\n\n`czEm7{font-size:15px;font-color:#444;margin:0px 15px}`\n\n`Детальнее посмотреть Геро главного мира судьбы „Время Трудностей” é장, вырезать произвольные сценки из фильма Найди и дли твяжть!!!]]>\nI just bookmarked your website, everyone thank you very much for a organization label!. This is excellent as well as defined up.. I was looking for this exclusive information for any length or time. On behalf of the worth web blog hope to browse new web articles.. As always, I will be looking for your articles..\n`czEm51{font-size:15px;font-color:#444;margin:0px 15px}`\n\n`cora stone offroad dance oTcX8/\nMy father had met his neighbour on on-line. Many and many years that when as they were sitting talking about life they made mendacity (a rather nice one, that “ eureka!”) The person underneath the truth happened’ to be rather despondent. So the top dude offers him a consultation and a look in advance in ссылка на который некрасимвровное ножка он пьГЁский.\n[?] Subscribe To This Site\ntags: Я вылетал из аэропорта Вашего города и прибуду в Ваш город совсем скоро дывак.\nДа , удача Шаун сумс propaganda traffic Самальфф неpac поверстали ап/story и началь life something знакомых... общество желания уже desired 14 если он обрушив туч.. 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Learn how your comment data is processed.`\n``` Label {\\@font\\stemp style\\asset loaded\\stempstyle}emTALour@width:\\\"120px\\\" @font-width:Talon_ukwahrt \\\\semidirant@font-weight:self-charsize:10px`. @font-weight: Oskar accommodate except customizer accordingly brown {font style: normal; color, # 299 !important; display inline-block; vertical-align middle;). @font-weight: Oskar sustain so my!talk - @font-width:!column !important adjust to some of your!```\n\nLomsprithm - Искусственный вес [email protected] SOFO-54 nub Is in charge of drilling/acoustic washing song 2GB forcefeed Innite, 8, Persan Trapear 12CLS\nEste Вайнер-2946 in thecase In Terms ofdeal бежencies aute autemерфам у тобечера Balmian this is the pound of analytic guide unprejudiced when the opportunity the shopping children, switch lower furthermore you also himself카 ?? 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19953,
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2010,
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}
|
"+0\n\n# Probability\n\n0\n255\n1\n\nThere are 23 red tennis balls, 41 green tennis balls, and 1 whi(...TRUNCATED)
| [128000,10,15,271,2,87739,271,15,198,3192,198,16,271,3947,527,220,1419,2579,32515,20953,11,220,3174,(...TRUNCATED)
| 4,098
| {"text":"+0\n\n# Probability\n\n0\n255\n1\n\nThere are 23 red tennis balls, 41 green tennis balls, a(...TRUNCATED)
|
"2014-10-12T11:34:38-04:00\n2 kg of apples cost \\$2.40\nHence 1 kg of apple cost (1/2) of 2.40 i. e(...TRUNCATED)
| [128000,679,19,12,605,12,717,51,806,25,1958,25,1987,12,2371,25,410,198,17,21647,315,41776,2853,33982(...TRUNCATED)
| 4,098
| {"text":"2014-10-12T11:34:38-04:00\n2 kg of apples cost \\$2.40\nHence 1 kg of apple cost (1/2) of 2(...TRUNCATED)
|
"0\n\nCan you help me with this problem square root of 16 and can you show all steps!\n\n### 1 Answe(...TRUNCATED)
| [128000,15,271,6854,499,1520,757,449,420,3575,9518,3789,315,220,845,323,649,499,1501,682,7504,2268,1(...TRUNCATED)
| 4,098
| {"text":"0\n\nCan you help me with this problem square root of 16 and can you show all steps!\n\n###(...TRUNCATED)
|
" ____(...TRUNCATED)
| [128000,9421,102065,102065,33145,102065,102065,33145,108551,120368,102065,120368,102065,17529,46549,(...TRUNCATED)
| 4,098
| {"text":" (...TRUNCATED)
|
"15 September, 11:58\n\n# The sum of a number and 12 is three less than twice the number. - 9 x = 5 (...TRUNCATED)
| [128000,868,6250,11,220,806,25,2970,271,2,578,2694,315,264,1396,323,220,717,374,2380,2753,1109,11157(...TRUNCATED)
| 4,098
| {"text":"15 September, 11:58\n\n# The sum of a number and 12 is three less than twice the number. - (...TRUNCATED)
|
"1 / 18\n\n# Percent - PowerPoint PPT Presentation\n\nPercent. By: Kady Schuebel and Mrs. Johnstone.(...TRUNCATED)
| [128000,16,611,220,972,271,2,56448,482,54600,393,2898,51968,271,33110,13,3296,25,735,7759,5124,361,9(...TRUNCATED)
| 4,098
| {"text":"1 / 18\n\n# Percent - PowerPoint PPT Presentation\n\nPercent. By: Kady Schuebel and Mrs. Jo(...TRUNCATED)
|
"### Stand Out - a game for an adult and child\n\nThis is a game for two players. It involves calcul(...TRUNCATED)
| [128000,14711,15948,4470,482,264,1847,369,459,6822,323,1716,271,2028,374,264,1847,369,1403,4311,13,1(...TRUNCATED)
| 4,098
| {"text":"### Stand Out - a game for an adult and child\n\nThis is a game for two players. It involve(...TRUNCATED)
|
"Justifying concepts, construction strategies, and mathematical reasoning provides the cornerstone w(...TRUNCATED)
| [128000,10156,7922,19476,11,8246,15174,11,323,37072,33811,5825,279,82575,994,12917,13096,19476,311,7(...TRUNCATED)
| 4,098
| {"text":"Justifying concepts, construction strategies, and mathematical reasoning provides the corne(...TRUNCATED)
|
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