Plane Trigonometry - Mathematics

Plane Trigonometry - Mathematics

A Treatise on Plane and Advanced Trigonometry (Dover Books on Mathematics) 7th Edition

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Crafton Hills College Summer 2021 Mathematics

Course Information

Course Name : MATH-103

Course Title : Plane Trigonometry

Course Description : Study of the circular functions, DeMoivre's Theorem and applications. Emphasis is placed on mastering trigonometric identities and the solution of trigonometric equations. If purchasing a used book, new software may need to be purchased at an additional expense.

Additional Course Information

Credit Type : Earned units for this course are applicable to an Associate Degree.

Transferability : Course credit transfers to CSU.

Prerequisite : MATH 095 or eligibility for MATH 103 as determined through the Crafton Hills College assessment process.

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Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. [1] He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions. [2] Euclid never used numbers to measure length, angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane.

This section is solely concerned with planes embedded in three dimensions: specifically, in R 3 .

Determination by contained points and lines Edit

In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following:

  • Three non-collinear points (points not on a single line).
  • A line and a point not on that line.
  • Two distinct but intersecting lines.
  • Two distinct but parallel lines.

Properties Edit

The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues:

  • Two distinct planes are either parallel or they intersect in a line.
  • A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
  • Two distinct lines perpendicular to the same plane must be parallel to each other.
  • Two distinct planes perpendicular to the same line must be parallel to each other.

Point–normal form and general form of the equation of a plane Edit

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let r0 be the position vector of some point P0 = (x0, y0, z0) , and let n = (a, b, c) be a nonzero vector. The plane determined by the point P0 and the vector n consists of those points P , with position vector r , such that the vector drawn from P0 to P is perpendicular to n . Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that

The dot here means a dot (scalar) product.
Expanded this becomes

which is the point–normal form of the equation of a plane. [3] This is just a linear equation

In mathematics it is a common convention to express the normal as a unit vector, but the above argument holds for a normal vector of any non-zero length.

Conversely, it is easily shown that if a, b, c and d are constants and a, b , and c are not all zero, then the graph of the equation

is a plane having the vector n = (a, b, c) as a normal. [4] This familiar equation for a plane is called the general form of the equation of the plane. [5]

Thus for example a regression equation of the form y = d + ax + cz (with b = −1 ) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Describing a plane with a point and two vectors lying on it Edit

Alternatively, a plane may be described parametrically as the set of all points of the form

where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, and r0 is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane. The vectors v and w can be perpendicular, but cannot be parallel.

Describing a plane through three points Edit

Method 1 Edit

The plane passing through p1 , p2 , and p3 can be described as the set of all points (x,y,z) that satisfy the following determinant equations:

Method 2 Edit

To describe the plane by an equation of the form a x + b y + c z + d = 0 , solve the following system of equations:

This system can be solved using Cramer's rule and basic matrix manipulations. Let

If D is non-zero (so for planes not through the origin) the values for a, b and c can be calculated as follows:

These equations are parametric in d. Setting d equal to any non-zero number and substituting it into these equations will yield one solution set.

Method 3 Edit

This plane can also be described by the "point and a normal vector" prescription above. A suitable normal vector is given by the cross product

and the point r0 can be taken to be any of the given points p1 , p2 or p3 [6] (or any other point in the plane).

Distance from a point to a plane Edit

Another vector form for the equation of a plane, known as the Hesse normal form relies on the parameter D. This form is: [5]

Line–plane intersection Edit

In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line.

Line of intersection between two planes Edit

The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in space may be written as r = c 1 n 1 + c 2 n 2 + λ ( n 1 × n 2 ) >=c_<1><oldsymbol >_<1>+c_<2><oldsymbol >_<2>+lambda (<oldsymbol >_<1> imes <oldsymbol >_<2>)> , since < n 1 , n 2 , ( n 1 × n 2 ) >>_<1>,<oldsymbol >_<2>,(<oldsymbol >_<1> imes <oldsymbol >_<2>)>> is a basis. We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for c 1 > and c 2 > .

Dihedral angle Edit

In addition to its familiar geometric structure, with isomorphisms that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of abstraction. Each level of abstraction corresponds to a specific category.

At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.

The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability.

In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the complex plane and the major area of complex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.

In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation.

In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point). This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature.

Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. The latter possibility finds an application in the theory of special relativity in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space.)

The one-point compactification of the plane is homeomorphic to a sphere (see stereographic projection) the open disk is homeomorphic to a sphere with the "north pole" missing adding that point completes the (compact) sphere. The result of this compactification is a manifold referred to as the Riemann sphere or the complex projective line. The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map.

The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the hyperbolic plane such diffeomorphism is conformal, but for the Euclidean plane it is not.

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1 Answer 1

WLOG we can translate things around so that the base of our rectangular prism has vertex $(0, 0, 0), (a, 0, 0), (0, b, 0), (a, b, 0)$ and $(0, 0, 0)$ is the vertex with the smallest $z$ -coordinate where the plane cut the rectangular prism.

Also, WLOG we can assume our normal vector $n = egin p q r end$ has length $|n| = 1$ .

With this setup, all we need to do is to find the other 2 vertices $(a, 0, z_1), (0, b, z_2)$ where the plane cut the rectangular prism. Since the cross section is a parallelogram, the area is then simply the length of the cross product of the position vectors of the 2 vertices.

This is the hint part. You can stop reading now and try to solve it yourself.

Since $gamma$ is the angle between the $k = egin 0 0 1 end$ and $n$ , by the property of dot product, $cos gamma = |k||n| cos gamma = k cdot n = egin 0 0 1 end cdot egin p q r end = r$

From now on, we can write $n = egin p q cos gamma end$

Therefore $p^2 + q^2 + cos^2 gamma = 1$ and hence $p^2 + q^2 = 1 - cos^2 gamma = sin^2 gamma$

Since the plane cut the rectangular prism at $(0, 0, 0)$ , it passes throught the origin. Hence we can write its equation as $egin p q cos gamma end cdot egin x y z end = 0$ which is the same as $px + qy + (cos gamma)z = 0$ .

As the plane cut through the rectangular prism at vertice $(a, 0, z_1)$ , the vertice satisfies the equation of the plane. Therefore, $p a + q 0 + (cos gamma)z_1 = 0$ and hence $z_1 = -frac$

Similarly, at vertice $(0, b, z_2)$ , we have $p 0 + q b + (cos gamma)z_2 = 0$ and hence $z_2 = -frac$

With these two $z$ -coordinates, we now know the position vectors of the 2 vertices are $egin a 0 -frac end$ and $egin 0 b -frac end$ .

Since $(0, 0, 0)$ is the vertex with the smallest $z$ -coordinate where the plane cut the rectangular prism, these 2 position vectors are the sides of the parallelogram.

[PDF] Download SL Loney Plane Trigonometry and Coordinate Geometry book

These books are great for learning concepts as well as to practice. Some topics are beyond JEE scope. It is a highly recommended book for those JEE aspirants who are preparing by themselves or facing difficulty in creating problem solving approach towards the subject.

Plane Trigonometry by SL LoneyBuy NowDownload PDF Now
The Elements of Coordinate geometry by SL LoneyBuy NowDownload PDF Now
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Plane trigonometry

Sidney Luxton Loney, M.A. (16 March 1860 – 16 May 1939) was a Professor of Mathematics at the Royal Holloway College, Egham, Surrey, and a fellow of Sidney Sussex College, Cambridge. He authored a number of mathematics texts, some of which have been reprinted numerous times. He is known as an early influence on Srinivasa Ramanujan.

Loney was educated at Maidstone Grammar School, in Tonbridge and at Sidney Luxton Loney, M.A. (16 March 1860 – 16 May 1939) was a Professor of Mathematics at the Royal Holloway College, Egham, Surrey, and a fellow of Sidney Sussex College, Cambridge. He authored a number of mathematics texts, some of which have been reprinted numerous times. He is known as an early influence on Srinivasa Ramanujan.

Loney was educated at Maidstone Grammar School, in Tonbridge and at Sidney Sussex College, Cambridge, where he graduated BA as 3rd Wrangler in 1882.


In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it looks like a flat infinite sheet of paper. There are several definitions of the plane. They are equivalent in the sense of Euclidean geometry, but they can be extended in different ways to define objects in other areas of mathematics. The only two-dimensional figure in our three-dimensional world is a shadow.

In some areas of mathematics, such as plane geometry or 2D computer graphics, the whole space in which the work is carried out is a single plane. In such situations, the definite article is used: the plane. Many fundamental tasks in geometry, trigonometry and graphing are performed in the two dimensional space, or in other words, in the plane.

A plane is a surface such that, given any three distinct points on the surface, the surface also contains all of the straight lines that pass through any two of them. One can introduce a Cartesian coordinate system on a given plane in order to label every point on it with a unique ordered pair, which is composed of two numbers and is the coordinate of the point.

Within any Euclidean space, a plane is uniquely determined by any of the following combinations:

3 Answers 3

I started my career as an archaeologist before I ended up in mathematics. I say this in order to emphasize that I am interested in trying to understand how people thought in the past&mdashthe usual "text" that an archaeologist reads is the collection of artifacts which are left behind, but there is also a very active field called Historical Archaeology which seeks to associate historical records with a "ground truth". From the point of view an "archaeologist" (or historian) of mathematics, I think that texts such as Loney may be interesting, and well worth reading. However, I would recommend against using such a text in an introductory class.

The cons which you mention are significant, but there are another couple of issues which should give you further pause:

Looking over the table of contents, it appears that much of the focus is on computation. For example, starting around page 106, there are many pages spent on computing the sines and cosines of angles using the angle sum and half-angle formulae. My experience is that modern exposition concerns itself more with the formulae themselves (as these recur in calculus), and largely elides explicit computation. Such computation is, perhaps, useful as an exercise, but a CAS can typically do the job faster and more accurately. In general, my preference would be to use a book which places much less emphasis on computation or which emphasizes the way in which modern computers can aid computation.

There are a lot of topics in that text which are kind of archaic, or which are wholly inappropriate for a modern precalculus class. For example, most of Chapters X and XI are not relevant in a modern classroom (there is no reason for students to be taught how to read a log table, for example). Much of Chapter XV seems to focus on aspects of geometry which, for better or worse, are not typically part of the standard US precalculus curriculum (some of it might show up in the high school curriculum, but a lot of it isn't part any standard curriculum prior to upper division courses in geometry (or, perhaps, math competition prep courses)). These topics certainly have some interest, but if the goal is to prepare students for a standard calculus curriculum, then they don't do any favors to the students.

And then there is Part II. Almost nothing in Part II is part of the precalculus curriculum (and nothing in Part II is mentioned in the course description in the question). It starts with series representations of the logarithm and exponential (though using notation which is difficult to parse by modern standards&mdashthe funky factorial and the lack of Sigma notation, for example), moves on to a couple of limits, then jumps into complex analysis. The entire second half of the book is, in most modern classrooms, covered in courses on calculus and complex analysis. It doesn't belong in a precalculus class.

The fact that Part II is inappropriate for a precalculus class isn't a big deal&mdashPart I certainly contains enough material for a semester&mdashbut I think that it might be better to select a book which is more narrowly focused on the topics which you actually need to cover.

In the question, it is noted that the language "may sound archaic to some students". I think that this fails to capture the magnitude of the problem&mdashI think that students are likely to bounce hard off of a topic which is presented with unfamiliar notation (and, by the way, unfamiliar notation which they won't ever see again) written in a form of English which is distinctly old fashioned. They inconsistent typesetting also does the book no favors (but now I'm being catty).

I am struggling for an analogy which is not hyperbolic, but I think that the following works: you don't teach students Russian by asking them to read Война и Мир (War and Peace) or Евгений Онегин (a novel by Pushkin) in the original pre-reform language right out of the gate. Get your students to read something from the 20th or 21st Century, first (maybe something like Один День Ивана Денисовича&mdashthe language is modern and pretty accessible, and it is still a classic). Anyone who wants to become a scholar of Russian literature should probably get familiar with pre-reform works eventually, but that isn't the place to start. Start with modern Russian, and work back.

Similarly, students of mathematics should start with a modern presentation and then, if they want to study the history of mathematics, start trying to tackle the classics (presuming that Loney is, indeed, a "classic", and not simply "old").

Basically, my recommendation would be to find another text. I don't think that Loney is appropriate for a modern introductory audience. At best, you might use it supplement the course (for example, one of my critiques of the text, above, is that if over-emphasizes computation, at least by modern standards however, there appear to be a lot of exercises in the text, which might prove useful). Moreover, there is nothing wrong with reading a book such a Loney's and drawing inspiration from it (personally, I have gotten a lot of milage in my precalc classes out of Gelfand's Method of Coordinates and Klein's Elementary Mathematics from an Higher Standpoint).

Unfortunately, I also don't have a lot of advice about which book you should use. There are a lot of books out there with titles like Precalculus (with Trigonometry!) and Trigonometry for the Precalculus Student and whatnot. Most of these books are 1,000 pages long, weigh 10 lbs., and cost $200+. They make good doorstops, but are otherwise too dilute and broad to be of much use. I'm also not a huge fan of the Schaum's Outlines as course texts. They are useful for exercises, but leave a lot to be desired vis-à-vis exposition (of course, that is kind of the point&mdashthey are outlines).

Perhaps consider one of the open source texts that are out there? For example, the OpenStax Algebra and Trigonometry or the Open Textbook Library's Trigonometry. I suspect that these books will suffer from many of the same problems as the $200+ tomes, but at least they are free. :

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