Trigonometric Ratios

5 Key excerpts on "Trigonometric Ratios"

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  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

  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

  Pre-Calculus Workbook For Dummies

  • Mary Jane Sterling(Author)
  • 2019(Publication Date)
  • For Dummies

    (Publisher)

  Then you see how to apply that knowledge to the unit circle, a very useful tool for graphically representing Trigonometric Ratios and relationships. From there, you can solve trig equations. Finally, these concepts are combined so that you can apply them to arcs. The ancient Greeks didn’t know what they started with trigonometry, but the modern applications are endless! Finding the Six Trigonometric Ratios In geometry, angles are measured in degrees, with describing a full circle on a coordinate plane. And another measure for angles is radians. Radians, from the word radius, are usually designated without a symbol for units; the radian measure is understood. Because both radians and degrees are used often in pre-calculus, you see both in use here. To convert radians to degrees and vice versa, you use the fact that radians, or. Therefore, to convert degrees to radians, just remember this proportion and substitute in what you know:. The represents an angle measured in degrees, and the represents an angle measured in radians. When solving right triangles or finding all the sides and angles, it’s important to remember the six basic trigonometric functions:,,,,, and. And there’s also the reciprocal relationships between the functions:,, and. One of the most famous acronyms in math is SOHCAHTOA. It helps you remember the first three Trigonometric Ratios: Q. Given ∆KLM in the following figure, find. A.. Because, you first need to find the hypotenuse. To do this, you need to use the Pythagorean Theorem, which says that. Using this, you find that, which becomes, so the hypotenuse is. Plugging this into your sine ratio, you get. You can rationalize the denominator and get. Q. Solve ∆RST, referring to the following figure. A.. Remember, solving a triangle means finding the measures of all the angles and sides. So you start by using the Pythagorean Theorem to find the hypotenuse:,. Next, use any trigonometric ratio to find an angle. You can use

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  eBook - ePub

  Mathematician's Delight

  • W. W. Sawyer(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  The formula given above thus becomes:

  It is because of this relation between 3,4, and 5 that the triangle is right-angled. Another such triangle is 5, 12, 13. 52 + 122 = 132 . If we draw an angle whose cosine is its sine will be

  Here we have the answer to the question raised in Chapter 2: the proof given above is, in essentials, that given in Euclid. The result is usually known as Pythagoras’ Theorem, and can be stated: if a, b , c are the lengths of the sides of a right-angled triangle, then a 2 + b 2 = c 2 . This result is essentially the same as the result we have just found. For if t is the angle between the sides a and c , a = c cos t and b = c sin t . (This result is obtained by enlarging the scale of our standard triangle, OPQ, c times.) So a 2 + b 2 = (c cos t )2 + (c sin t )2 . This last expression is equal to c 2 multiplied by (cos t )2 + (sin t )2 . By the result proved above, this is the same as c 2 multiplied by 1: that is, c 2 . So a 2 + b 2 = c 2 follows from our earlier result, by simple algebra.

  The Cosine Formula

  So far we have considered only triangles containing a right-angle. We will now consider a more general problem. Suppose we have a triangle ABC, and we know the lengths AB and AC, and the size of the angle BAC. How long is BC ? (It might be impossible to measure BC directly, owing to mountains, rivers, swamps, etc.)

  In books on trigonometry it is usual to write a, b, c for the lengths of the sides BC, CA, AB and to write A, B, C for the three angles of the triangle. Thus, a is the side opposite the angle A, etc. Our problem is: given b, c, A, to find a.

  Can this problem be solved at all? Are the facts given sufficient to allow us to draw a plan of ABC

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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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