Trigonometric Functions of General Angles

5 Key excerpts on "Trigonometric Functions of General Angles"

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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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  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

  Trigonometry For Dummies

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

    (Publisher)

  T he six basic trig functions all had humble beginnings with the right triangle and its angles. The unit circle opens up a whole new world for the input values into those functions. Because of the nature of trig functions — they repeat the same patterns over and over — the output values show up regularly. This repetition is a good thing; you recognize where in the pattern a particular input belongs and then assign the output. Life is good.

  Defining Trig Functions for All Angles

  So many angles are used in trigonometry and other math areas, and the majority of those angles are multiples of 30 and 45 degrees. So, having a trick up your sleeve letting you quickly access the function values of this frequent-flier list of angles makes perfect sense. All you need to know or memorize are the values of the trig functions for 0-, 30-, 45-, 60-, and 90-degree angles in order to determine all the trig functions of all the angles, positive or negative, that are multiples of 30 or 45 degrees, which are the two most basic, foundational angles. Finding these function values for a particular angle is a three-step process: (1) Find the measure of the angle's reference angle, (2) Assign the correct numerical value, and (3) Determine whether the function value is positive or negative.

  Putting reference angles to use

  The first step to finding the function value of one of the angles that's a multiple of 30 or 45 degrees is to find the reference angle. When the reference angle comes out to be 0, 30, 45, 60, or 90 degrees, you can use the function value of that angle and then figure out the sign (see the next section). Use Table 8-1 or Table 8-2 to find the reference angle.

  All angles with a 30-degree reference angle have trig functions whose absolute values are the same as those of the 30-degree angle. The sines of 30, 150, 210, and 330 degrees, for example, are all either or . Likewise, using a 45-degree angle as a reference angle, the cosines of 45, 135, 225, and 315 degrees, for example, are all or

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  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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  Engineering Mathematics

  A Programmed Approach, 3th Edition

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

    (Publisher)

  Fig. 3.4 ).

  The other circular functions, known as tangent, cotangent, secant and cosecant, can be defined in terms of cosine and sine. In fact

  tan     θ = 

  sin   θ

  cos   θ

          cot θ  =

  cos θ

  sin   θ

  sec   θ = 

  1

  cos   θ

            cosec   θ = 

  1

  sin   θ

  However, whereas cosine and sine have the real numbers ℝ as their domain, these subsidiary functions are not defined for all real numbers. □  Obtain the domain of each of these subsidiary circular functions by using the convention of the maximal domain.

  a  tan θ is defined whenever cos θ ≠ 0. From Fig. 3.4 we see that this is when θ is not an odd multiple of π/2. Any odd number can be written in the form 2n + 1 where n ∈ ℤ. Therefore the domain of the tangent function is

  A =

  {

  x |

  x     ∈     ℝ ,     x   ≠

  (

  2 n + 1

  )

  π / 2

  , n ∈ ℤ

  }

  b  cot θ is defined whenever sin θ ≠ 0. So θ must not be a multiple of π. Therefore the domain of the cotangent function is

  A =

  {

  x |

  x     ∈     ℝ ,     x   ≠ n π , n ∈ ℤ

  }

  Fig. 3.5 The tangent function.

  Fig. 3.6 The cosecant function.

  The domains of the secant and cosecant are the same as those of the tangent and cotangent respectively. ■

  The graph of y = tan x shows that the tangent function has period π (Fig. 3.5 ).

  The graph of the sine, cosine and tangent functions can be used to draw the graphs of the cosecant (Fig. 3.6 ), secant (Fig. 3.7 ) and cotangent functions. The graph of y = sec x has the same shape as the graph of y = cosec x. To obtain the graph of y = cosec x from the graph of y = sec x we merely need to relocate the y-axis through x = π/2 and relabel.

  Fig. 3.7 The secant function.

  You will remember that we write cos

  n

  θ instead of (cos θ)

  n

  when n is a natural number. This must not be confused with cos (nθ), and you should be alert to the fact that this notation does not hold good when n

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