80. To determine the tangent of twice or thrice a given arc in functions of the arc and radius. substituting the value of tan 2a, and reducing, we have, 81. To find the sine of half an arc in terms of the functions By formula (2) of the arc and radius. cos 2a 1 - 2 sin2 a. For a, substitute a, and we have, cos α = 1 2 sin2 a; 82. To find the cosine of half a given arc in terms of the. functions of the arc and radius. 83. To find the tangent of half a given arc, in functions of the arc and radius. Divide formula (0) by (p), and we have, Multiplying both terms of the second member by √1-cos a Multiplying both terms by the denominator √1 + cos a 84. The formulas of Articles 71, 72, 73, 74, furnish a great number of consequences; among which it will be enough to mention those of most frequent use. By adding and subtracting we obtain the four which follow, sin (a + b) + sin (a - b) = 2 sin a cos b, (~) cos (a + b) + cos (a - b) cos (a - b) cos (a + b) 2 sin a sin b, and which serve to change a product of several sines or cosines into linear sines or cosines, that is, into sines and cosines multiplied only by constant quantities. 85. If in these formulas we put a + b = p, a − b 2 g1 Я 2sin(p + q) cos } (p + q) ⚫ These formulas are the algebraic enunciations of so Inany theorems. The first expresses that, the sum of the sines of two arcs is to the difference of those sines, as the tangent of half the sum of the arcs is to the tangent of half their difference. HOMOGENEITY OF TERMS. 87. An expression is said to be homogeneous, when each of its terms contains the same number of literal factors. Thus, is homogeneous, since each term contains two literal factors. This equation merely expresses the numerical relation between the values of sin a, cos2 a, and unity. If we pass from the radius 1 to any other radius, as R, it becomes necessary to replace these abstract numbers by their corres ponding literal factors. the radius of a circle For this, we must observe, that bears the same ratio to any one of the in which the sin a, in the first member, is calculated to the radius 1, and in the second, to the radius R. If, now, we substitute this value of sin a to radius 1, in equation (2), we have, an expression which is homogeneous: and any expression may be made homogeneous in the same manner; or, it may be made so, by simply multiplying each term by such a power of R as shall give the same number of linear factors in all the terms. 88. Since the sine of an arc divided by the radius is equal to the sine of another arc containing an equal num. ber of degrees divided by its radius, we may, if we please, define the sine of an arc to be the ratio of the radius to the perpendicular let fall from one extremity of the arc on a diameter passing through the other extremity. Giving similar definitions to the other functions of the arc, each will have a corresponding function in either angle of a triangle. For, if in a right angled triangle, we let A = right angle; B = angle at base; C = vertical angle; a = hypothenuse; c = base; b= perpendicular, we may deduce all the functions of the angle without any reference to the circle. For, let us call, by definition, Each of these expressions, regarded as a ratio, is a mere abstract number. If we make the hypothenuse a = 1, the abstract numbers will then represent parts of a rightangled triangle, or the corresponding functions of a circle whose radius is unity. Formulas relating to Triangles. 89. Let ACB be any triangle, and designate the sides by the letters a, b, c; then (Art. 21), sin A a sin A ; sin C C sin C с that is, the sines of the angles are to each other as their opposite sides. that is, the sum of any two sides is to their difference, as the tangent of half the sum of the opposite angles to the tangent of half their difference. 91. In case of a right-angled triangle, in which the right angle is B, we have (Art. 24), 1 : tan A :: C : a; hence, a = c tan A, . (2) And again (Art. 25), 1 : cos A :: b : c; hence, cb cos A, . (3) |