Trigonometric Functions

6 Key excerpts on "Trigonometric Functions"

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  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 5 Trigonometry and Trigonometric Functions 5.1 Introduction

  The word trigonometry is derived from two Greek words, together meaning measuring the sides of a triangle . The subject was originally developed to solve geometric problems involving triangles. One of its uses lies in determining heights and distances , which are not easy to measure otherwise. It has been very useful in surveying , navigation , and astronomy . Applications have now further widened.

  At school level, in geometry, we have studied the definitions of trigonometric ratios of acute angles in terms of the ratios of sides of a right-angled triangle.

  Note that in the right-angled triangle OAR , if the lengths of the sides are respectively denoted by B (for base), P (for perpendicular), and H (for hypotenuse), as shown in Figure 5.1 , then the angle θ (in degrees) is an acute angle (i.e., 0° < θ < 90°). It is for such angle(s) that we have defined trigonometric ratios in earlier classes.1

  Figure 5.1 Right angled triangle defining trigonometric ratios.

  Now, in our study of trigonometry, it is required to extend the notion of an angle in such a way that its measure can be of any magnitude and sign . Once this is done, the trigonometric ratios are defined for angles of all magnitudes and sign. Finally, by identifying these magnitudes and signs of angles, with real numbers, we say that the trigonometric ratios of directed angles represent Trigonometric Functions of real variables

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  eBook - ePub

  The Ellipse

  A Historical and Mathematical Journey

  • Arthur Mazer(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  y ) coordinates.

  Figure 5.2 Trigonometric Functions as coordinates of a unit circle.

  The angle θ is sometimes expressed in radians and sometimes expressed in degrees. It is worth one’s effort to be able to use both units of measurement.

  5.1.2 Triangles

  For angles between 0˚ and 90˚ (from 0 to π /2radians), the Trigonometric Functions correspond to ratios of right triangles as illustrated in Figure 5.3 .

  Figure 5.3 Trigonometric Functions as triangular ratios.

  The equivalence between the definitions in the preceding two tables is seen by noting the triangle formed between a point on the unit circle, the origin, and the point along the x axis given by the x coordinate of the original point. The ratios that define the Trigonometric Functions are the same for all similar triangles.

  5.1.3 Examples

  Using the definitions, it is possible to determine the Trigonometric Functions for some values of θ . Examples are given below. These examples are the first entries into a trigonometric table that is further developed in subsequent sections.

  Example 5.1

  Determine the trigonometric for the value θ = π /4 rad (45˚).

  Solution

  When the angle θ is π/4 rad, x = y along the unit circle (see Figure 5.4 ). With the assistance of the Pythagorean theorem, the values for x and y

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  eBook - ePub

  Elementary Calculus

  An Infinitesimal Approach

  • H. Jerome Keisler(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  7

  Trigonometric Functions

  7.1 TRIGONOMETRY

  In this chapter we shall study the Trigonometric Functions, i.e., the sine and cosine function and other functions that are built up from them. Let us start from the beginning and introduce the basic concepts of trigonometry.

  The unit circle x2 + y2 = 1 has radius 1 and center at the origin.

  Two points P and Q on the unit circle determine an arc , an angle ∠POQ, and a sector POQ. The arc starts at P and goes counterclockwise to Q along the circle. The sector POQ is the region bounded by the arc and the lines OP and OQ. As Figure 7.1.1 shows, the arcs and are different.

  Figure 7.1.1

  Trigonometry is based on the notion of the length of an arc. Lengths of curves were introduced in Section 6.3 . Although that section provides a useful background, this chapter can also be studied independently of Chapter 6 . As a starting point we shall give a formula for the length of an arc in terms of the area of a sector. (This formula was proved as a theorem in Section 6.3 but can also be taken as the definition of arc length.)

  DEFINITION

  The length of an arc on the unit circle is equal to twice the area of the sector POQ, s = 2A.

  This formula can be seen intuitively as follows. Consider a small arc of length ∆s (Figure 7.1.2 ). The sector POQ is a thin wedge which is almost a right triangle of altitude one and base ∆s. Thus . Making ∆s infinitesimal and adding up, we get .

  The number π ~ 3.14159 is defined as the area of the unit circle. Thus the unit circle has circumference 2π.

  The area of a sector POQ is a definite integral. For example, if P is the point P(1, 0) and the point Q(x, y) is in the first quadrant, then we see from Figure 7.1.3 that the area is

  Notice that A(x) is a continuous function of x

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  eBook - ePub

  Trigonometry For Dummies

  • Mary Jane Sterling(Author)
  • 2014(Publication Date)
  • For Dummies

    (Publisher)

  . This answer is undefined, which means that angle γ has no secant. For the reason that γ has no secant, refer to the next section.

  Defining Domains and Ranges of Trig Functions

  The domain of a function consists of all the input values that a function can handle — the way the function is defined. Of course, you want to get output values (which make up the range ) when you enter input values (for the basics on domain and range, see Chapter 3 ). But sometimes, when you input something that doesn't belong in the function, you end up with some impossible situations. In these cases, you need to limit what you put into the function — the domain has to be restricted. For example, the cosecant is defined as the hypotenuse divided by the opposite side (see Chapter 7 ). If the terminal side of the angle is on the x- axis, then the opposite side is 0, and you're asked to divide by 0. Impossible!

  Trig functions have domains that are angle measures (the inputs are all angles), either in degrees or radians. The outputs of the trig functions are real numbers. The hitch here is that the different trig functions have different domains and ranges. You can't put just any angle into some of the functions. Sine and cosine are very cooperative and have the same domain and range. The tangent function and the reciprocal functions, however, all differ. The best way to describe these different domains and ranges is visually: Refer to the coordinate plane with a circle centered at the origin and a right triangle inside it, formed by dropping a line from any point (x,y ) on the circle to the x- axis (see Figure 9-7 ). Remember that r

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  eBook - ePub

  AS and A Level Maths For Dummies

  • Colin Beveridge(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Chapter 12

  Taking Trigonometry Further

  In This Chapter

  Sketching basic trig functions

  Using trig identities and compound angle formulas

  Finding multiple solutions

  Completing trig proofs

  As you might expect from a maths qualification designed to take your understanding further, the trigonometry you did at GCSE gets a lot more involved at A level. In some respects, it gets simpler, as you begin to see how everything links together, but in most respects, it’s messy and a little confusing. (You can handle it, though. I believe in you!)

  In A level trigonometry, my advice is that radians are the correct way to measure angles, unless you’re explicitly told otherwise (for example, if the question asks for an angle to the nearest degree or tells you ). This chapter uses radians almost exclusively (the full rant explaining why is in Chapter 11 ).

  Here, I show you how to exploit symmetry to draw awesome graphs of the trig functions and how to use right-angled triangles to remember your identities. You get to grips with compound angles and finding all the solutions to trigonometric equations, and you discover how to prove things on demand.

  Sketching Up Symmetries

  A little secret: I love the basic trig functions. I love the way you can reflect and rotate them without changing their nature. I love the way that differentiating sine and cosine gives you variations on sine and cosine. I love that you can express them in terms of e , which you won’t see unless you do Further Maths and/or university maths. I love the way that adding them together gives you another variation on sine and cosine. I love their link with Pythagoras. There’s just the right balance of change and constancy for my taste, so I award a Lifetime Achievement Award for being a brilliant function, and is in the running for Best Supporting Role.

  The symmetries are the best bit, though. Sketching the graphs of the Trigonometric Functions is by far the most obvious way to see which symmetries you can use. These little drawings will come in very useful in later sections!

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  eBook - ePub

  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  Trigonometry and geometry 3

  In the last two chapters we described some of the basic terminology which we need. We also picked up a few techniques which should prove useful later on. Soon we shall begin to develop the differential calculus, but before we do that we must make sure that we can handle any geometrical or trigonometrical problem that arises.

  After working through this chapter you should be able to □  Use circular functions, recognize their graphs and be able to determine their domains; □  Solve equations involving circular functions; □  Recognize the equations of standard geometrical curves; □  Transform equations involving polar coordinates into those involving cartesian coordinates. At the end of this chapter we shall solve practical problems in surveying and in circuits.

  This chapter contains background work, and so it is possible that much of it will be familiar to you. If this is the case, then it is best to regard it as revision material. We shall be reviewing work on elementary trigonometry and coordinate geometry. If any section is very well known to you then simply read it through and devote your attention to that which is less familiar.

  3.1      COORDINATE SYSTEMS

  You are probably quite familiar with the cartesian coordinate system. In this system every point in the plane is determined uniquely by an ordered pair of numbers (x, y). To do this, two fixed straight lines are laid at right angles to one another; these are called the x-axis and the y-axis. Their point of intersection is represented by O and is called the origin (Fig. 3.1 ). The quadrants so formed are labelled anticlockwise as the first quadrant, second quadrant, third quadrant and fourth quadrant respectively.

  Fig. 3.1 The cartesian system.

  Given any point P, the absolute values of x and y are then obtained from the shortest distance of P to the y-axis and the x

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