Sine and Cosine Rules

5 Key excerpts on "Sine and Cosine Rules"

• 

  eBook - ePub

  Mathematics for Scientific and Technical Students

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

    (Publisher)

  Fig. 3.19 into two right-angled triangles.

  The sine rule states that,

  The cosine rule states that

  Fig. 3.19

  In any triangle there are six unknowns: three sides and three angles. In order to be able to determine all six values at least three of these must be known. The cosine rule is used if the following are known:

  (i) 3 sides;

  (ii) 2 sides and the angle in between them.

  The sine rule is used if the following are known

  (i) 1 side and 2 angles;

  (ii) 2 sides and an angle which is not in between.

  Note: At least one side must always be specified. The application of the two rules is shown in Examples 3.8 to 3.11 .

  Example 3.8 Find the length of the base of the template shown in Fig. 3.20 .

  Fig. 3.20

  Since one side and two angles are known the sine rule is used. Now angle C = 180- (65 + 75) = 40° (because the angles of a triangle always total 180°)

  Example 3.9  Fig. 3.21 shows an assembly for a crane. Find the angle between the girders AB and BC .

  Fig. 3.21

  Two sides and an angle not between them are known. Therefore the sine rule is used. For ease of working we use the inverted form of the sine rule.

  Referring to the Fig. 3.21 , the angle ABC is greater than 90°. This other solution, following Example 3.6 , is in the second quadrant, that is 180 - 66.8 = 113.2° which is 110° correct to two significant figures, that is B = 110°.

  Example 3.10  The total current I taken by an a.c. circuit is given by the line BC in Fig. 3.22 . Calculate the value of this total current and the phase angle φ.

  Fig. 3.22

  Two sides and the angle in between are known so that the cosine rule is used.

  Now that four quantities are known either rule may be used to find φ. Using the inverted sine rule

  Example 3.11  Fig. 3.23a

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  eBook - ePub

  Trigonometry For Dummies

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

    (Publisher)

  . The situation with the ratios is the same as with the sine function — the values are going to be less than or equal to 1 (the latter only when your triangle is a single segment or when dealing with circles), never greater than 1, because the hypotenuse is the denominator.

  The two ratios for the cosine are the same as those for the sine — except the angles are reversed. This property is true of the sines and cosines of complementary angles in a right triangle (meaning those angles that add up to 90 degrees).

  If θ and λ are the two acute angles of a right triangle, then sin θ = cos λ and cos θ = sin λ .

  Now for an example. To find the cosine of angle β in a right triangle if the two legs are each feet in length:

  1. Find the length of the hypotenuse.
     Using the Pythagorean theorem, a 2 + b 2 = c 2 (see Chapter 6 ), and replacing both a and b with the given measure, solve for c .
     The hypotenuse is feet long.
  2. Use the ratio for cosine, adjacent over hypotenuse, to find the answer.

  The tangent function: Opposite over adjacent

  The third trig function, tangent, is abbreviated tan. This function uses just the measures of the two legs and doesn't use the hypotenuse at all. The tangent is described with this ratio: . No restriction or rule on the respective sizes of these sides exists — the opposite side can be larger, or the adjacent side can be larger. So, the tangent ratio produces numbers that are very large, very small, and everything in between. If you hike on back to Figure 7-2 , you see that the tangents are and . And in case you're wondering whether the two tangents of the acute angles are always reciprocals (flips) of one another, the answer is yes. The trig identities in Chapter 11

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  eBook - ePub

  Flash 3D

  Animation, Interactivity, and Games

  • Jim Ver Hague, Chris Jackson(Authors)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  Figure 5.12 , let dx represent the difference between the x-axis values of the two points and let dy represent the difference between the y-axis values of the two points. Then dx = x2 − x1 and dy = y2 − y1. The distance between the two points forms the hypotenuse of a right triangle with dx and dy as the other two sides.

  Figure 5.12 Calculating the distance between two points

  From the Pythagorean theorem we know that distance2 = dx2 + dy2 or

  In ActionScript, the square root is a defined function that we can use for just such occasions as this. The equation for the distance expression above would be written as shown in Figure 5.13 .

  Figure 5.13 Formula for the distance between two points

  The Trig Functions

  Sine, cosine, and tangent are known as the trig functions. All they do is represent the three ratios of the sides of a right triangle in relation to the angles in the triangle. Referring to Figure 5.14 , the three trig functions are defined as follows:

  Figure 5.14 Definition of the trig functions

  In English we say that the sine of the angle labeled angle is equal to the side opposite from angle divided by the hypotenuse. The cosine of angle is equal to the side adjacent (but not the hypotenuse) to angle divided by the hypotenuse. The tangent of angle is equal to the opposite side divided by the adjacent side.

  In an earlier discussion, it was pointed out that the sum of the interior angles of all triangles is equal to 180 degrees. When we have a right triangle, the two angles that are not the right angle must then add up to 90 degrees. Again referring to Figure 5.14 above, we see that

  sin(angle) = cos(90 − angle) cos(angle) = sin(90 − angle)

  In other words, the sine of one angle in a right triangle is the cosine of the other angle and vice versa. Later when we graph these functions, we will see that they essentially create the same curves.

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  eBook - ePub

  Spherical Geometry and Its Applications

  • Marshall A. Whittlesey(Author)
  • 2019(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 4

  Trigonometry

  15 Spherical Pythagorean theorem and law of sines

  Spherical trigonometry is the key to the study of the precise relationship among distances and angles on the sphere. Central to this is the relationship among the distances and angles in a triangle. But what makes these relationships different from the corresponding relationships in the plane?

  The central geometric difference between the plane and the sphere is, of course, the fact that the sphere is “curved.” In the plane, if we take two rays with the same endpoint, then the rays separate away from the common endpoint. On the sphere, two such rays separate for a while away from the endpoint but then converge back together at the antipode of the endpoint. We first quantify this behavior of rays on the sphere with a proposition.

  Proposition 15.1 (A-12) Suppose two rays form an angle with measure θ. Let x be the spherical distance between two points (one on each ray) at spherical distance ϕ from the vertex. Then sin(

  x 2

  ) = sin(ϕ)sin(

  θ 2

  ).

  We have seen this proposition already: it is Proposition 6.1 of §6. (See also Figure 2.8.) There we proved it as a theorem for spheres in space, using the properties of three-dimensional geometry. In our axiomatic system we do not have those techniques available and we will assume it as an axiom. However, this is not strictly necessary: if we assume certain properties of trigonometric functions and certain properties of the real numbers from real analysis, we can prove Proposition 15.1 as a theorem. This is sufficiently difficult that it is beyond the scope of this book. However, we refer the interested reader to [Ca1916], p. 136 , or [Wo1945], to see how this can be done.1

  We use this result to determine properties of right triangles on the sphere. We adopt the following notation. In a triangle Δ

  s

  ABC, let A, B, and C denote the measures of the angles at vertices A, B, and C, respectively, and let a, b, and c denote the lengths of the sides opposite vertices A, B, and C, respectively. This means that the letters A, B, and C

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  eBook - ePub

  Mathematics for Engineering

  • W Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  Figure 11.1 , we can write

  AB2 + BC2 = AC2

  If we divide both sides of the equation by AC2 we obtain:

  But AB/AC = sin θ and BC/AC = cos θ. Hence we have the very commonly used relationship of

  sin2 θ + cos2 θ = 1

  (sin θ)2 is written as sin2 θ and (cos θ)2 as cos2 θ.

  Figure 11.1 Pythagoras theorem

  11.2.1 Further relationships

  We can obtain two further relationships from the above equation. If we divide the sin2 θ + cos2 θ = 1 equation by cos2 θ we obtain:

  and so, since sin θ/cos θ = tan θ and 1/cos θ = sec θ, we obtain the relationship:

  tan2 θ + 1 = sec2 θ

  If we divide the sin2 θ + cos2 θ = 1 equation by sin2 θ we obtain:

  and so, since cos θ/sin θ = cot θ and 1/sin θ = cosec θ, we obtain:

  1 + cot2 θ = cosec2 θ

  Example

  Show that

  We can write the expression as:

  Example

  Show that

  We can write the equation with a common denominator as: Multiplying out the brackets gives:

  As sin2 θ + cos2 θ = 1, then this can be written as:

  Revision

  1    Simplify the following:

  (a) , (b) , (c) , (d) sinθ cosθ tan θ.

  2    Show that .

  3    Show that tan θ + cot θ = sec θ cosec θ.

  4    Show that 1 - 2 sin2 θ = 2 cos2 θ - 1.

  5    Show that .

  11.3 Trigonometric ratios of sums of angles

  It is often useful to express the trigonometric ratios of angles such as A + B or A - B in terms of the trigonometric ratios of A and B. Consider the two right-angled triangles OPQ and OQR shown in Figure 11.2 :

  Figure 11.2 Compound angle

  Hence:

  sin(A + B) = sin A cos B + cos A sin B[1]

  If we replace B by -B we obtain:

  sin(A - B) = sin A cos B - cos A sin B  [2]

  If in equation [1] we replace A by (π/2 - A) we obtain:

  cos(A + B) = cos A cos B - sin A sin B[3]

  If in equation [3] we replace B by -B we obtain:

  cos(A - B) = cos A cos B + sin A sin B[4]

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