each second into 60 thirds, &c., marked with the characters &c. Thus, 32° 24′ 13′′ 22′′ is 32 degrees, 24 minutes, 13 seconds, 22 thirds.* A degree, then, is not a magnitude of a given length; but a certain portion of the whole circumference of any circle. It is evident, that the 360th part of a large circle is greater than the same part of a small one. On the other hand, the number of degrees in a small circle, is the same as in a large one. The fourth part of a circle is called a quadrant, and contains 90 degrees. 74. To measure an angle, a cirele is so described that its center shall be the angular point, and its periphery shall cut the two lines which include the angle. The arc between the two lines is considered a measure of the angle, because, by Euc. 33. 6, angles at the center of a given circle, have the same ratio to each other, as the arcs on which they stand. Thus the arc AB, (Fig. 2.) is a measure of the angle ACB. It is immaterial what is the size of the circle, provided it cuts the lines which include the angle. Thus, the angle ACD (Fig. 4.) is measured by either of the arcs AG, ag. For ACD is to ACH, as AG to AH, or as ag to ah. (Euc. 33. 6.) 75. In the circle ADGH, (Fig. 2.) let the two diameters AG and DH be perpendicular to each other. The angles ACD, DCG, GCH, and HCA, will be right angles; and the periphery of the circle will be divided into four equal parts, each containing 90 degrees. As a right angle is subtended by an arc of 90°, the angle itself is said to contain 90°. Hence, in two right angles, there are 180°; in four right angles, 360°; and in any other angle, as many degrees, as in the arc by which it is subtended. 76. The sum of the three angles of any triangle being equal to two right angles, (Euć. 32. 1.) is equal to 180°. Hence, there can never be more than one obtuse angle in a triangle. For the sum of two obtuse angles is more than 180°. 77. The COMPLEMENT of an arc or an angle, is the difference between the arc or angle and 90 degrees. The complement of the arc AB (Fig. 2.) is DB; and the complement of the angle ACB is DCB. The complement of the arc BDG is also DB. *See note E. The complement of 10° is 80°, of 20° is 70°, of 60° is 30°, of 120° is 30°, of 170° is 80°, &c. Hence, an acute angle and its complement are always equal to 90°. The angles ACB and DCB are together equal to a right angle. The two acute angles of a right angled triangle are equal to 90°: therefore each is the complement of the other. 78. The SUPPLEMENT of an arc or an angle is the difference between the arc or angle and 180 degrees. The supplement of the arc BDG (Fig. 2.) is AB; and the supplement of the angle BCG is BCÀ. The supplement of 10° is 170°, of 80° is 100°, of 120° is 60°, of 150° is 30°, &c. Hence an angle and its supplement are always equal to 180°. The angles BCA and BCG are together equal to two right angles. 79. Cor. As the three angles of a plane triangle are equal to two right angles, that is, to 180° (Euc. 32. 1.) the sum of any two of them is the supplement of the other. So that the third angle may be found, by subtracting the sum of the other two from 180°. Or the sum of any two may be found, by subtracting the third from 180°. 80. A straight line drawn from the centre of a circle to any part of the periphery, is called a radius of the circle. In many calculations, it is convenient to consider the radius, whatever be its length, as a unit, (Alg. 510.) To this must be referred the numbers expressing the lengths of other lines. Thus, 20 will be twenty times the radius, and 0.75, three fourths of the radius. Definitions of Sines, Tangents, Secants, &c. 81. To facilitate the calculations in trigonometry, there are drawn, within and about the circle, a number of straight lines, called Sines, Tangents, Secants, &c. With these the learner should make himself perfectly familiar. 82, The SINE of an arc is a straight line drawn from one end of the arc, perpendicular to a diameter which passes through the other end. Thus, BG (Fig. 3.) is the sine of the arc AG. For BG is a line drawn from the end G of the arc, perpendicular to the diameter AM which passes through the other end A of the arc. Cor. The sine is half the chord of double the arc. The sine BG is half PG, which is the chord of the arc PAG, double the arc AG. 83. The VERSED SINE of an arc is that part of the diameter which is between the sine and the arc. Thus, BA is the versed sine of the arc AG. 84. The TANGENT of an arc, is a straight line drawn perpendicular from the extremity of the diameter which passes through one end of the arc, and extended till it meets a line drawn from the center through the other end. Thus, AD (Fig. 3.) is the tangent of the arc AG. 85. The SECANT of an arc, is a straight line drawn from the center, through one end of the arc, and extended to the tangent which is drawn from the other end. Thus, CD (Fig. 3.) is the secant of the arc AG. 86. In Trigonometry, the terms tangent and secant have a more limited meaning, than in Geometry. In both, indeed, the tangent touches the circle, and the secant cuts it. But in Geometry, these lines are of no determinate length; whereas, in Trigonometry, they extend from the diameter to the point in which they intersect each other. 87. The lines just defined are sines, tangents, and secants of arcs. BG (Fig. 3.) is the sine of the arc AG. But this arc subtends the angle GCA. BG is then the sine of the are which subtends the angle GCA. This is more concisely expressed, by saying that BG is the sine of the angle GCA. And universally, the sine, tangent, and secant of an arc, are said to be the sine, tangent, and secant of the angle which stands at the center of the circle, and is subtended by the arc. Whenever, therefore, the sine, tangent, or secant of an angle is spoken of; we are to suppose a circle to be drawn whose center is the angular point; and that the lines mentioned belong to that arc of the periphery which subtends the angle. 88. The sine and tangent of an acute angle, are opposite to the angle. But the secant is one of the lines which include the angle. Thus, the sine BG, and the tangent AD, (Fig. 3.) are opposite to the angle DCA. But the secant CD is one of the lines which include the angle, 89. The sine complement or COSINE of an angle, is the sine of the COMPLEMENT of that angle. Thus, if the diameter HO (Fig. 3.) be perpendicular to MA, the angle HCG is the complement of ACG; (Art. 77.) and LG, or its equal CB, is the sine of HCG. (Art. 82.) It is, therefore, the cosine of GCA. On the other hand, GB is the sine of GCA, and the cosine of GCH. So also the cotangent of an angle is the tangent of the complement of the angle. Thus, HF is the cotangent of GCA. And the cosecant of an angle is the secant of the complement of the angle. Thus, CF is the cosecant of GCA. Hence, as in a right angled triangle, one of the acute angles is the complement of the other; (Art. 77.) the sine, tangent, and secant of one of these anglès, are the cosine, co-tangent, and cosecant of the other. 90. The sine, tangent, and secant of the supplement of an angle, are each equal to the sine, tangent, and secant of the angle itself. It will be seen, by applying the definition (Art. 82.) to the figure, that the sine of the obtuse angle GCM is BG, which is also the sine of the acute angle GCA. It should be observed, however, that the sine of an acute angle is opposite to it; while the sine of an obtuse angle falls without the angle, and is opposite to its supplement. Thus BG, the sine of the angle MCG, is not opposite to MCG, but to its supplement ACG. The tangent of the obtuse angle MCG is MT, or its equal AD, which is also the tangent of ACG. And the secant of MCG is CD, which is also the secant of ACG. 91. But the versed sine of an angle is not the same as that of its supplement. The versed sine of an acute angle is equal to the difference between the cosine and radius. But the versed sine of an obtuse angle is equal to the sum of the cosine and radius. Thus, the versed sine of ACG is AB=AC -BC. (Art. 83.) But the versed sine of MCG is MB=MC +BC. Relations of Sines, Tangents, Secants, &c., to each other. 92. The relations of the sine, tangent, secant, cosine, &c., to each other, are easily derived from the proportions of the sides of similar triangles. (Euc. 4. 6.) In the quadrant ACH, (Fig. 3.) these lines form three similar triangles, viz. ACD, BCG or LCG, and HCF. For, in each of these, there is one right angle, because the sines and tangents are, by definition, perpendicular to AC; as the cosine and cotangent are to CH. The lines CH, BG, and AD, are parallel, because CA makes a right angle with each. (Euc. 27. 1.) For the same reason, CA, LG, and HF, are parallel. The alternate angles GCL, BGC, and the opposite angle CDA, are equal; (Euc. 29. 1.) as are also the angles GCB, LGC, and HFC. The triangles ACD, BCG, and HCF, are therefore similar. It should also be observed, that the line BC, between the sine and the center of the circle, is parallel and equal to the cosine; and that LC, between the cosine and center, is parallel and equal to the sine; (Euc. 34. 1.) so that one may be taken for the other, in any calculation. 93. From these similar triangles, are derived the following proportions; in which R is put for radius, By comparing the triangles CLG and CHF, 4. CH: CL:: HF: LG, that is, R sin:: cot: cos. R cosec : cos; cot. sin R R: cosec, Therefore R2=sin x cosec, By comparing the triangles CAD and CHF, 7. CH: AD: : CF: CD, that is, R tan: : cosec: sec. 8. CA HF:: CD: CF 9. AD AC:: CH: HF R cot sec: cosec. tan R R cot. Therefore R2=tan X cot. It will not be necessary for the learner to commit these proportions to memory. But he ought to make himself so familiar with the manner of stating them from the figure, as to be able to explain them, whenever they are referred to. |