Page:Encyclopædia Britannica, Ninth Edition, v. 23.djvu/584

that $n\pi +\alpha$ includes all the angles which have the same tangent as $\alpha$.

From the Pythagorean theorem, the sum of the squares of the projections of any straight line upon two straight lines at right tween angles to one another is equal to the square on the projected line, we get $\sin ^{2}\alpha +\cos ^{2}\alpha =1$, and from this by the help of the definitions of the other functions we deduce the relations $1+\tan ^{2}\alpha ={\text{sec}}^{2}\alpha$, $1+\cot ^{2}\alpha ={\text{cosec}}^{2}\alpha$. We have now six relations between the six functions ; these enable us to express any five of these functions in terms of the sixth. The following table shows the values of the trigonometrical functions of the angles 0, ${\frac {1}{2}}\pi$, $\pi$, ${\frac {3}{2}}\pi$, $2\pi$, and the signs of the functions of angles between these values; I denotes numerical increase and D numerical decrease.

Angle	$0$	$0\dots {\frac {1}{2}}\pi$	${\frac {1}{2}}\pi$	${\frac {1}{2}}\pi \dots \pi$	$\pi$	$\pi \dots {\frac {3}{2}}\pi$	${\frac {3}{2}}\pi$	${\frac {3}{2}}\pi \dots 2\pi$	$2\pi$
Sine	0	+I	1	+D	0	-I	-1	-D	0
Cosine	1	+D	0	-I	-1	-D	0	+I	1
Tangent	0	+I	$\pm \infty$	-D	0	+I	$\pm \infty$	-D	0
Cotangent	$\pm \infty$	+D	0	-I	$\pm \infty$	+D	0	-I	$\pm \infty$
Secant	1	+I	$\pm \infty$	-D	-1	-I	$\pm \infty$	+D	1
Cosecant	$\pm \infty$	+D	1	+I	$\pm \infty$	-D	-1	-I	$\pm \infty$

The correctness of the table may be verified from the figure by considering the magnitudes of the projections of OP for different positions.

Values of trigonometrical functions for some angles. The following table shows the sine and cosine of some angles for which the values of the functions may be obtained geometrically:—

		sine	cosine
${\frac {\pi }{12}}$	15°	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	75°	${\frac {5}{12}}\pi$
${\frac {\pi }{10}}$	18°	${\frac {{\sqrt {5}}-1}{4}}$	${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$	72°	${\frac {2}{5}}\pi$
${\frac {\pi }{6}}$	80°	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	60°	${\frac {1}{3}}\pi$
${\frac {\pi }{5}}$	36°	${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$	${\frac {{\sqrt {5}}+1}{4}}$	54	${\frac {3}{10}}\pi$
${\frac {\pi }{4}}$	45°	${\frac {1}{\sqrt {2}}}$	${\frac {1}{\sqrt {2}}}$	45°	${\frac {1}{4}}\pi$
		cosine	sine

These are obtained as follows. (1) ${\frac {\pi }{4}}$. The sine and cosine of this angle are equal to one another, since $\sin {\frac {\pi }{4}}=\cos({\frac {\pi }{2}}-{\frac {\pi }{4}})$, and since the sum of the squares of the sine and cosine is unity each is ${\frac {1}{\sqrt {2}}}$. (2) ${\frac {\pi }{6}}$ and ${\frac {\pi }{3}}$. Consider an equilateral triangle; the projection of one side on another is obviously half a side; hence the cosine of an angle of the triangle is ${\frac {1}{2}}$ or $\cos {\frac {\pi }{3}}={\frac {1}{2}}$, and from this the sine is found. (3) ${\frac {\pi }{10}},{\frac {\pi }{5}},{\frac {2\pi }{5}},{\frac {3\pi }{10}}$. In the triangle constructed in Euc. iv., each angle at the base is ${\frac {2\pi }{5}}$, and the vertical angle is ${\frac {\pi }{5}}$/. If a be a side and b be the base, we have by the construction $a(a-b)=b^{2}$; hence $2b=a({\sqrt {5}}-1)$; the sine of ${\frac {\pi }{10}}$ is ${\frac {b}{2a}}$ or ${\frac {{\sqrt {5}}-1}{4}}$, and $\cos {\frac {\pi }{5}}$ is ${\frac {a}{2b}}={\frac {{\sqrt {5}}+1}{4}}$. (4) ${\frac {\pi }{12}},{\frac {5\pi }{12}}$. Consider a right-angled triangle, having an angle ${\frac {\pi }{6}}$. Bisect this angle, then the opposite side is cut by the bisector in the ratio of ${\sqrt {3}}$ to 2; hence the length of the smaller segment is to that of the whole in the ratio of ${\sqrt {3}}$ to ${\sqrt {3}}+2$, therefore $\tan {\frac {1}{12}}\pi ={\frac {\sqrt {3}}{{\sqrt {3}}+2}}\tan {\frac {1}{6}}\pi$ or $\tan {\frac {1}{12}}\pi =2-{\sqrt {3}}$, and from this we can obtain $\sin {\frac {1}{12}}\pi$ and $\cos {\frac {1}{12}}\pi$.

Formulæ for some and cosine of sum and difference of two angles. Draw a straight line OD making any angle A with a fixed straight line OA, and draw OF making an angle B with OD, this angle being measured positively in the same direction as A ; draw FE a perpendicular on DO (produced if necessary). The projection of OF on OA is the sum of the projections of OE and EF on OA. Now OE is the projection of OF on DO, and is therefore equal to OF cos B, and EF is the projection of OF on a straight line making an angle + JT with OD, and is therefore equal to O^sin B ; hence OF cos (A +)= OEcos A+EFcos (ir+A) OF (cos A cos B - sin A sin B or cos (A +B)= cos A cos B- sin A sin B.

The angles A, B are absolutely unrestricted in magnitude, and thus this formula is perfectly general. We may change the sign of B, thus cos (A B) = cos A cos ( B) sin A sin ( - B or cos (A -) = cos A cos B + sin A sin B.

If we projected the sides of the triangle OEF on a straight line making an angle +^ir with OA we should obtain the formulas

= sin A cosJ5cos A sin B,

which are really contained in the cosine formula, since we may put frr - B for B. The formulæ

/^_LT> tan^4tan.B ,/>i DN cot ^4 ta.n(A) = - l^tan^tan^ ~ cot^?cot^

are immediately deducible from the above formulas. The equations

sin C+ sin D=2 sin (G+D) cosfc (G- D), sin C f -sin2)=2sini(C -Z>)cos|(C +Z)),\\ cosD + cos C=2cosl(G+D)cosl(C-D), cos D- cos G=2sin^(C+D)sin^(G-D),

may be obtained directly by the method of projections. Take two equal straight lines OG, OD, making angles G, D with OA, and draw OE perpendicular to CD. The angle which OE makes

with OA is ^(G+D) and that which DC makes the angle COE is (G-D). The sum of the projections of OD and DE on OA is equal to that of OE, and the sum of the projections of OD and DE is equal to that of OC ; hence the sum of the projections of 00 and OD is twice that of OE, or cos G + cos D = 2 cos ^(G+D) cos (C-D). The difference of the projections of OD and OG Fi S- 4 - on OA is equal to that of ED, hence we have the formula cos D - cos G=2 sin %(G+D) sin ^(C- D). The other two formulas will be obtained by projecting on a straight line inclined at an angle + ir to OA.

Sum of series of cosines in arithmetical progression. As another example of the use of projections, we will find the sum of the series cos a + cos (o + /3) + cos (a + 2/3) + . . . + cos (a + ?i-l |8). series of Suppose an unclosed polygon each angle of which is TT - /3 to be in- scribed in a circle, and let A lt A 2 , A 3 , . . ., A n be n + 1 consecutive angular points ; let D be the diameter of the circle ; and suppose a straight line drawn making an angle a with AA lt then a + /3, progres- a + 2/3, . . . are the angles it makes with A 1 A 2 , A 2 A 3 , . . . ; we have by s i u - projections (1 /3 a -) j-*C j = ^^ 1 (cos o + cos o + /3 + . .". -f cos a + n - 1 ft), J^lgO AA 2 2 hence the sum of the series of cosines is cos ( a H ^ -

Formulæ for some and cosine of sum of angles By a double application of the addition formulas we may obtain the formulas

sin (A l + A 2 + A 3 ) = sin A cos A% cos A 3 + cos A l sin A 2 cos A 3 + cos A l cos y* 2 sin ^ 3 - sin A l sin ^ 2 sin A 3 ; cos (-<4 1 + A 2 + A 3 ) = cos ^j cos A 2 cos ^4 3 cos A sin ^4 2 sin A 3 - sin ^4 X cos A 2 sin ^ 3 - sin A l sin ^ 2 cos ^ 3 .

We can by induction extend these formulas to the case of n angles. Assume sin (A 1 +A 2 + ... + A n ) = S 1 -S 3 + S 5 - .. . cos (A-i + A 2 + . . . + A n ) = SQ S 2 + 84 ... where S r denotes the sum of the products of the sines of? of the angles and the cosines of the remaining n-r angles ; then we have

sin (A 1 + A 2 + ... + A n + A n+1 ) = cos A n+l (S l S 3 + S 6 . . . ) + sin A n+ i(S -S 2 + S t - ...).

The right-hand side of this equation may be written (S 1 cos A n+1 + S sin A n+l ) - (S 3 cos A n+l + S 2 sin A n+1 ) + ..., or S 1 -S 3 + ... where S r denotes the quantity which corresponds for n + l angles to S r for n angles ; similarly we may proceed with the cosine for mula. The theorems are true for n=2 and n=3 ; thus they are true generally. The formulas cos 2 A = cos- A - sin 2 A = 2 cos z A -1 = 1-2 sin 2 ^, sin 2A = 2 sin A cos A, tan 2A = - , sin 5 A = 3 sin A - 4 sin 3 A, cos 2 A = 4 cos 3 ^ - 3 cos A, sin nA = , . . . 1 ^ sm A n(n- s n ~ 3 A sin 3 A + . .. Formulas for mul tiple and sub-mul tiple angles. + (-1) r n(n-l)...(n-2r) 2r + l coa n ~ 2r ~ l A cos nA = cos nA cos"~ 2 ^ sin? A + ...

may all be deduced from the addition formula?, by making the angles all equal. From the last two formulas we obtain by division tannA ntan ^- 3 tan A

In the particular case of n= 3 we have tan ZA = -i _ o t a~s