Plane trigonometry, by S.L. Loney.

470

TRIGONOMETRY.

Put y =-, and the equation is
n
3 3 3n 2z + n3= 0.................................(1).
3 1
Now cos 0- cos 0 -- cos 30=0.......................(2).
4 4
Equations (1) and (2) are the same if
1 1
z=cos0, n2=-, and — os30=-n3,
4
i.e.if n-2'
and cos 30= - = cos 120~.......................(3).
The roots of (3) are clearly
40~, 40 + 120~, and 40 + 240~,
so that z=cos 40~, or cos 160~, or cos 280~... y=2 cos 40~, or 2 cos 160~, or 2 cos 280~... x=y-2= -2+2cos40~, or -2-2cos20~, or -2+2cos80~.
On referring to the tables we then have the values of x.
EXAMPLES. LXVII.
Solve the equations
1. 2x3-3x-1=0. 2. x3+3x2-1=0. 3. x3-24x-32=0,
4. x3-6x2+6x+8=0. 5. x3-21x+7=0.
6. x3+4x2+2x-1=0. 7. x3-7x+5=0.
Maximum and Minimum Values.
399. In Art. 133 we have given one example of the
maximum value of a trigonometrical expression.
We add another example.
If x and y be two positive angles whose sum is a constant
angle a( v7r), find when sin x sin y is a maximum, and
extend the theorem to more than two angles.