Triangle Rules

2 Key excerpts on "Triangle Rules"

• 

  eBook - ePub

  Mathematics for Scientific and Technical Students

  • H. Davies, H.G. Davies, G.A. Hicks(Authors)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  Chapter 3

  Trigonometry

     

  3.1 Introduction

  Trigonometry is the branch of Mathematics that deals with the relationships between the sides and the angles of a triangle. Provided that a minimum of

  (i) 3 sides,

  (ii) 2 sides and 1 angle or

  (iii) 1 side and 2 angles

  are known about any triangle, the other unknown sides or angles can be calculated using trigonometrical methods.

  Trigonometry is based on the trigonometrical ratios of sine, cosine and tangent, which are the ratios between the sides of a right-angled triangle. The values of these ratios depend upon the size of the angles, and do not depend on the size of the triangle.

  3.2 Trigonometric ratios

  In Fig. 3.1 sides are given names with respect to the angle θ. The side opposite the right-angle (90°) is called the hypotenuse. The other two sides are named according to their position relative to the angle. The ratios are defined as

  For a known angle the values of these ratios can be obtained from an electronic calculator. For example sin 40°=0.6428 can be obtained by entering 40 and pressing the sin key.

  Fig. 3.1

  Example 3.1  Find the lengths of the unknown sides in the steel bracket ABC shown in Fig 3.2 .

  Fig. 3.2

  Using     with θ = 60°   and  

  Multiply both sides by 21.4:

  Using     again with θ = 60°   and   h = 21.4cm

  Example 3.2 Fig 3.3 shows a voltage diagram for an electronic circuit. Calculate the phase angle ϕ between the two voltages.

  Fig. 3.3

  Note: ϕ = cos-1 0.5965 means that ϕ is an angle which has a cos of 0.5965. The value of ϕ can be obtained from the calculator by entering 0.5965 and pressing the

  cos−1

  key. cos-1 is the inverse of the cosine (see Section 3.15 ).

  3.3 Theorem of Pythagoras

  This is an important theorem and is a useful alternative method of finding the third side of a right-angled triangle when the other two sides are known. In the right-angled triangle in Fig 3.4 the sides are labelled a, b, c according to the angles that they are opposite.

  Fig. 3.4

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  eBook - ePub

  Geometry For Dummies

  • Mark Ryan(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Good point. The Pythagorean Theorem is easier for some triangles (especially if you’re allowed to use your calculator). But — take my word for it — this triple triangle technique can come in handy. Take your pick. Getting to Know Two Special Right Triangles Make sure you know the two right triangles in this section: the triangle and the triangle. They come up in many, many geometry problems, not to mention their frequent appearance in trigonometry, pre-calculus, and calculus. Despite the pesky irrational (square-root) lengths they have for some of their sides, they’re both more basic and more important than the Pythagorean triple triangles I discuss earlier. They’re more basic because they’re the progeny of the square and equilateral triangle, and they’re more important because their angles are nice fractions of a right angle. The 45°- 45°- 90° triangle — half a square The triangle (or isosceles right triangle): The triangle is a triangle with angles of,, and and sides in the ratio of. Note that it’s the shape of half a square, cut along the square’s diagonal, and that it’s also an isosceles triangle (both legs have the same length). See Figure 8-7. © John Wiley & Sons, Inc. FIGURE 8-7: The triangle. Try a couple of problems. Find the lengths of the unknown sides in triangles BAT and BOY shown in Figure 8-8. © John Wiley & Sons, Inc. FIGURE 8-8: Find the missing lengths. You can solve triangle problems in two ways: the formal book method and the street-smart method. Try ’em both and take your pick. The formal method uses the ratio of the sides from Figure 8-7. For, because one of the legs is 8, the x in the ratio is 8

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