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The hypothenuse is that side which is opposite to the rightangle.
The base and perpendicular are distinguished by reference to the angle in question :
i.e. the base is adjacent to the angle in question, and the perp.is opposite to the angle in question.
Thus, in reference to the angle MPO,
MO, being opposite to the angle MPO, is its perp.: and PM, being adjacent to the angle MPO, is its base.
71. Names have been given to the 6 ratios of the sides of the right-angled triangle as follows. perp.
base 1. Sine
2. Co-sine = hyp.
hyp. 5. Secant
6. Co-secant base
perp. 72. The following observations will aid the student in learning these definitions.
(1) We will call the Sine, Tangent, and Secant the Primary Ratios; the Cosine, Cotangent, and Cosecant the Secondary Ratios. (2) Note, then, that each
P Secondary Ratio is derived from the corresponding Primary Ratio by interchanging Base and Perpendicular.
(3) In the triangle OMP, call each side by the little letter corresponding to the angle opposite it. Then, we follow the same order of lettering in the
M Primary Ratios as in the Sides.
73. Just as the lines base, hyp., and perp. are defined in reference to a certain angle, so the names of the ratios must be understood as defined in reference to that angle.
If A denotes any angle, we speak of the sine of A, the cosine of A, &c. These expressions are abbreviated into sin A, cos A, &c.
If A represents the angle MOP in the preceding figures, the following will be the more complete definitions of these ratios : MP
OM 1. Sin A
2. Cos A
OM 3. Tan A
4. Cot A
OP 5. Sec A
6. Cox A*OM
74. We have shown that for every determinate value of the acute angle A, there is a determinate value of each of the ratios of the pairs of sides of the right-angled triangle containing A; and conversely.
Hence sin A, cos A, &c. are what are called Functions of A : i.e. quantities which depend on A and vary as A varies. They are also single-valued functions of A, i.e. they have one determinate value for every determinate value of A.
75. The student must specially remember that
(1) These functions of angles, being ratios of a line to a line, are abstract numbers.
(2) Sin A is a compound symbol representing a single numerical quantity. It cannot be divided algebraically into sin and A. Thus, we cannot say 2 sin A = sin 2A, or sin (A + B) sin A + sin B, as if sin were a multiplier or factor.
We Usually written cosec A. In the abbreviations of the secondary ratios, the student should attend to the final letters s, t, x.
must treat sin A as one whole symbol, as if it were always enclosed in an impregnable bracket. (sin A), (sin A), &c., are, however, always written in the forms sinA, sin A, &c., in order that the bracket may be dispensed with in writing.
So of the other ratios.
76. Explanation of the prefix Co.
Now the side OM, which is base to 2 MOP, is perp. to 2 MPO, and the side MP, which is perp. to Z MOP, is base to – MPO.
.. the ratios of MPO will be obtained from those of MOP by interchanging base and perp. But this is equivalent to an interchange between the corresponding primary and secondary ratios. Hence,
cosin A = sin of complement of A,
= sec of complement of A. Since, similarly,
sin A = cosin of complement of A,
= cosec of complement of A, we may say
sin = co-cosin, tan = co-cotan, sec = co-cosec. The above results should be carefully worked out by a reference to the figure.
77. Since, by above, the cosine of any angle is the sine of some other angle, and so of the others, therefore, general propositions about the sine will be true of the cosine; and those about the tangent will be true of the cotangent; and those about the secant will be true of the cosecant. The next article will illustrate this statement.
Fundamental relations of Inequality between the ratios of any
one angle. 78. From the proposition that of the three sides of a right-angled triangle the hypothenuse is larger than either of the others, the following inequalities amongst the ratios of an angle may be deduced :
Since, sine and cosine have hypothenuse in denominator, sine and cosine are always less than unity.
Since secant and cosecant have hypothenuse in numerator, secant and cosecant are always greater than unity.
Since sine and cosine have hypothenuse in denominator, and have the same numerator as tangent and cotangent respectively, sine is less than tangent, and cosine is less than cotangent.
Since secant and cosecant have hypothenuse in numerator, and the same denominator as tangent and cotangent respectively,
secant is greater than tangent, and cosecant than cotangent.
Thus the following trios are in ascending order : sine, tangent, secant : cosine, cotangent, cosecant.
Fundamental relations of Equality between the ratios of any
hyp. base (3) sec A x cos A
base hyp. The triple or zigzag relations.
base hyp. (4) sin A x cot A x sec A
= 1. hyp. perp.
base (5) cos A x tan A x cox A
1. hyp. base
There are thus 3 independent dual relations, and 2 independent triple relations.
These 5 relations are not all independent. Any one of them may be deduced algebraically from the other four.
These equations follow at once from the definitions of the ratios, and do not at all depend on any property of the right-angled triangle.
80. The Squared or Pythagorean relations.
By the famous 47th Proposition of Euc. I., attributed by tradition to Pythagoras, we have, in any right-angled triangle, sum of squares on sides containing the right angle
square on side opposite the right angle. Thus, algebraically,
(Perp.) + (Base) =(Hyp.). To reduce this equation to ratio form, we may divide successively by (hyp.), (base)?, and (perp.). Thus Perp.
i.e. sino A + cos2 A = 1. Нур., Нур.,
i.e. tano A +1=sec? A.
Нур. . (8)
i.e. 1 + cot? A = cox? A. Perp.
Perp. There are thus 3 independent squared relations. Of the 8 relations given in this and the preceding article, any 5, including one at least of the squared relations, are independent; from these 5 the other 3 may be algebraically deduced.
These squared relations depend on the Pythagorean property of the right-angled triangle, being in fact merely transformed expressions of that property.
81. The relations of inequality of Art. 78, will help the student to remember the relations of equality of Arts. 79 and 80. Thus, the equation sino A + cos? A = 1, which may be written
sinA =1 - cos” A or cos? A = 1 - sino A, shows that sin A and cos A are each less than 1. The equation tan’ A + 1 = seco A, which may be written
tano A = seco A -1 or seco A – tanA = 1, shows that sec A is greater than 1 and greater than tan A.