and is therefore a maximum when a-B=0, or a=| Thus the maximum value of cos a cos B is cos2 Thus the maximum value of cos a+cos ẞ is 2 cos Similar theorems hold in case of the sine. 2° Example 1. If A, B, C are the angles of a triangle, find the maximum value of sin A+ sin B + sin C and of sin A sin B sin C. Let us suppose that C remains constant, while A and B vary. Hence, so long as any two of the angles A, B, C are unequal, the expression sin A+ sin B+ sin C is not a maximum; that is, the expression is a maximum when A=B=C=60°. 2 sin A sin B sin C={cos (A-B) – cos (A+B)} sin C ={cos (A−B)+cos C}sin C. This expression is a maximum when A=B. Hence, by reasoning as before, sin A sin B sin C has its maximum value when A=B=C=60°. Example 2. If a and ẞ are two angles, each lying between 0 and П whose sum is constant, find the minimum value of sec a+ sec ß. Since a +ẞ is constant, this expression is least when the denomi nators are greatest; that is, when a=ẞ= a+ß 2 320. If a, ß, y, d, ...... are n angles, each lying between 0 and whose sum is constant, to find the maximum value of π 2 Let cos a cos ẞ cos y cos d...... a+B+y+d+......=σ. Suppose that any two of the angles, say a and B, are unequal; then if in the given product we replace the two unequal factors a+ß and cos a+B 2' cos a and cosẞ by the two equal factors cos the value of the product is increased while the sum of the angles remains unaltered. Hence so long as any two of the angles a, B, y, d, ... are unequal the product is not a maximum; that is, the product is a maximum when all the angles are equal. In this case each angle= σ the maximum value of cos a+cos B+cos y+.. ......n cos n 321. The methods of solution used in the following examples are worthy of notice. Example 1. Shew that tan 3a cot a cannot lie between 3 and 1 3' These two fractional values of tan2 a must be positive, and therefore n must be greater than 3 or less than 1 3 Example 2. If a and b are positive quantities, of which a is the greater, find the minimum value of a sec – b tan 0. Denote the expression by x, and put tan 0=t; then x=a√√1+t2 - bt; .. b2t2+2bxt+x2=a2 (1+t2); .. t2 (b2 — a2)+2bxt + x2 - a2=0. In order that the values of t found from this equation may be real, Thus the minimum value is √√a2 – b2. Example 3. If a, b, c, k are constant quantities and a, ß, y variable quantities subject to the relation a tan a+b tan ẞ+c tany=k, find the minimum value of tan2 a+tan2 ß+tan2 y. By multiplying out and re-arranging the terms, we have (a2+b2+c2) (tan2 a +tan2 ß+tan2 y) − (a tan a + b tan ẞ+c tan y)2 =(btany - c tan ẞ)2 + (c tan a - a tan y)2+ (a tan ß – b tan a)2. But the minimum value of the right side of this equation is zero; hence the minimum value of (a2+b2+ c2) (tan2 a+tan2 ß+tan2 y) − k2=0; that is, the minimum value of tana+tan2 ß+tan2y= k2 constants of the equations. To determine this relation we eliminate 6, and the result is called the eliminant of the given equations. 323. The following examples will illustrate some useful methods of elimination. Example 1. Eliminate between the equations l cos 0+m sin 0+n=0 and p cos @ + q sin @ +r=0. From the given equations, we have by cross multiplication whence by squaring, adding, and clearing of fractions, we obtain The particular instance in which q=l and p= -m is of frequent occurrence in Analytical Geometry. In this case the eliminant may be written down at once; for we have 23. sin a+sin ẞ+sin y>sin (a+B+y). If a and b are two positive quantities of which a is the greater, shew that a cosec 0>b cot 0+√√ a2 - b2. 26. If a, b, c, k are constant positive quantities, and a, ß, y variable quantities subject to the relation a cos a+b cos ẞ+c cos y=k, find the minimum value of cos2 a+cos2 B+cos2y and of a cos2 a+b cos2 ß+c cos2 y. Elimination. 322. No general rules can be given for the elimination of some assigned quantity or quantities from two or more trigonometrical equations. The form of the equations will often suggest special methods, and in addition to the usual algebraical artifices we shall always have at our disposal the identical relations subsisting between the trigonometrical functions. Thus suppose it is required to eliminate from the equations From this example we see that since satisfies two equations (either of which is sufficient to determine independent of 0, which subsists between ) there is a relation, the coefficients and |