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Answers

$41-54$ : Terminal Points and Reference Numbers Find (a) thereference number for each value of $t$ and (b) the terminal point determined by $t$ .

$t=-\frac{11 \pi}{3}$

We have to highlight the point E which is three comma zero. We have the point to be, which is four common to why don't see is generally five comma food. Why entity? We have six common six point is three negative Full blind deaf iss three negative three point Z We have three minus two and point edge. We have three and minus one. So one by one will highlight these points So three comma zero x coordinate is three and y cornet zero. So this point is, well, ex quarter it is three units and white quarter to zero So this is pointy nexus point these four common toe So x quantities four And why corn it is too. So we should highlight this point. It is mine to be sees five drama for so excess five and wise Full toe Xs Five year and wise full corresponding point We'll get at this point this point is C de six comma 62 x coordinated six and white when it is also six. So we'll have this point This is the appointees three going on minus four Also export industry and why corn? It is minus four corresponding At this point, we have a good day. This point is he We have fine Deficit three Goma minus three. So three and minus three. We have this point, which is point if these three going on minus two. So three and minus two. We have this point. This is a and edges tree going on minus one. So excess three and wise minus one at this point. Is that so? I hope you understood. A B C D E f zh. All points are mentioned here.

We have to highlight the points point A is one comma one trying to be will have to common tree buoyancy is 35 point de we have four comma seven point is minus 21 blind If it's minus two, coma to point Z is minus off to common. Three point edges minus of two comma. Full point is we have minus two. Come on five And we have point Kate which is minus two coma six. So we'll highlight these points. Point days one comma 12 X quantities one And why quantities one. So one comma one. We are having this point. This is pointed point bees to go, Montri. So explain it is to why come according the streets or to common three. We have this point. This is point B. Seas three, Goma Fire So excess three and white slaves recording the The point is here did his point. C these four comma seven. So this is four. And why call it is seven soul. We have this point. Point B Next appointees minus two comma one. So minus two. Number one. It is pointy. If his minus two common to so minus two and So this is point death Zis minus two Elementary So minus two common tree. We have this point. This is the and the same time at U minus two comma food so minus two Common for this rich. So if he'll do this clearly minus two comma one is a This point is e minus Took all my two years f So this is if minus two Geometry is easy so we have this point c minus took on for his edge So we have this point edge minus two. Come on five is a So again, this point, we have the and minus two Common six is gay. So we have this point. Ok, I hope you undecided ABC D e f zh geeky. These all points are highlighted on this X Y axis coordinate system.

So a triangle is a right triangle if and only if one of its interior angles is, well, a right angle, which is 90 degrees. Well, that means that the vector components of any two sides of the triangle must be orthogonal, right? If I mean perpendicular is means or orthogonal basically mean the same thing. If we have vectors, we typically say orthogonal rather than perpendicular. But we need to have two sides of a triangle to be perpendicular or at a 90 degree angle, meaning that at least two other vectors former the sides must be orthogonal. We know that two vectors U and V If we have two vectors, that's a U and V are any two vectors then you nvr orthogonal if and only if the dot product you dotted with V is equal to zero. So here were given that these vergis is a B and C are vergis ease of a right of a triangle received right triangle off some triangle ABC. So we find the vector components off the sides A, B, B, C and C A. So to find the vector component of the side A. B um, well, the best components are obtained by subtracting the coordinates off the initial point from the coordinates of the terminal point So defined a B, the vector A B we just take this is gonna be equal to we take four minus three. So that's one. And then three minus zero. So three and then zero minus two. So negative two. Okay, so there is a B. Then we find the better components off the side BC. So to find the component of the side BC while we take eight minus four. So that's four. And then one minus three. So negative two and then negative one minus zero. So negative one. Okay, so there's a record components off the side BC. And then let's find the components of the size C eight. So there's a component here. The vector. See? A that's gonna be equal to three minus eight. So negative five and then zero minus one. So negative one and then one to minus a negative one. So that's two plus one, which is three. Okay, so now we find the dot product of these vectors, right? If any other that product or zero, then we have a right triangle. So, um Well, let's find a B died with BC. So if we do a b the vector A B and we got that with BC, or do we have this is gonna be equal to We have one times four plus three times Negative two and then plus, uh, negative. Two times negative one. Okay, well, this is equal to what, four minus six, uh, plus two, which is indeed equal to zero. So we have that the dot product a be down with B c is equal to zero. This means that a B is orthogonal or perpendicular orthogonal to B. C. Right? Therefore, since we have two factors that are orthogonal to each other, right, if any two factors are orthogonal, that means we have a right angle. Means we have a right triangle. So yes, we do. This triangle is yes, indeed A right triangle. So yes, right. We are a right triangle. Take care

Okay, Let us find the reference number and terminal point for A T equals two pi over three. OK, we know this point would be in the second quadrant. Therefore, we know that we would do pie minus two pi over three to determine the reference number. So if we do this math, we would figure out that the reference number is pi over three. In other words, this is the remainder. We're looking at intervals. Okay, now we're looking at the terminal Point is established before the terminal point is always co sign of tea. Com A sign of tea and we know what Artie is. Artie's too pie over three. So if I plug in directly into this, end up with co sign of two pi over three com, a sign of two pi or three which ultimately simplifies to negative 1/2 comma squirt of three over two. So this would be our terminal point over here. Negative. 1/2 calm. Escort of three over to and our reference numbers pi over three


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