TRIGONOMETRY. 461 TRIGONOMETRY. as MjBj, negative; the revolving radius OB, is press cos (270° — 6) in terms of a function of P. always positive. Thus (see Fig. 4) the signs of This angle is in the third ([uadrant anil therefore the functions of an angle not exceeding 00° are its cosine is negative. To make the angle 270° — $ all plus. The versine and coversine are evidently less than 90° we must suhtract 180° and we have cos (270° — 6) = — cos (00° — 6). But cos (90° — 6) = sin , and we have cos (270° — 0) = — sin e. The increasing of an angle by 360° or any mul- tiple of 300° does not alter the value of the trig- onometric functions of that angle. (See Func- tion.) It appears from the geometric representa- tion of the functions that the values of the sines and cosines of all real angles lie within the inter- val + 1, — 1 ; the values of the tangents and co- tangents of all real angles lie within the interval -|- 00 and — oo ; those of the secants and co- secants without the interval +1, — 1, as is show^n in the following table: Fig. 4. always positive. The sine and thf cosecant are positive in the first and second quadrants, and negative in the third and fourth. The cosine and the secant are positive in the first and foiirtli quadrants and negative in the second and third. The tangent and cotangent are positive in the first and third quadrants, and negative in the second and fourtli. We have, therefore, the following relations: sin ( — a) = — sin o cos ( — a) = cos o tan ( — o) ^ — tan a cot ( — o) = — cot o For the definition of a negative angle, see Angle. A given value of a function does not, however, uniquely determine the angle, as will appear from the following table of relations: FUNCTION 0° 90° 180° 270° 3G0O Sine
00 1 1
GO 1 1
oo 00
— »
— 1
— 1
CO — 1 — 1
00 —0
1
The variations of the functions are best ex- hibited by means of graphs. In the figures the arcs are laid off" as abscissas and the functions as ordinates. See Coordinates. GRAPH OF eiN $^ sin (00° cos (90° First Quadrant e) = cos aitan (90° - 6) = sin fli cot (90° - e) = cot e e) = tan e sin (180° — «) = cos (180° — 0) = tan(lvS0° — fl) = cot (180° — 9) = Second Quadrant sin (00° +e) = cos (90° + e) =
- 37t
- 2
in 2 cosS — sin — cot cot cot (90° + 9) = — tan sin - cos -tanff tan (00° 8): Third Quadrant GRAPH OF SEC^. sin (180° +«)=— sine cos (180° -(-») = — cos « tan (180° -i-9) = + tan« cot (180° +e) = 4-cot« sin (270° — 0) = cos (270° — ») = tan (270° — 9) = cot (270° —8) = — cos — sin cot tan Fourth Quadrant sin(.360°— 9) = - cos (360° — 9) = tan(.300° — 9) = - cot(360°— 9)=- -sin 8 cos 8 -tan 8 -cots sin (270° +9)= — cos 9 cos (270° +0)= sine tan (270° +0)=—cot8 cot (270° -f »)=- tanS Thus to express the trigonometric functions of angles greater than 90° in terms of those of angles less than 90°, determine first the sign of the function to be so expressed, next subtract from the angle whatever multiple of 00 is neces- sary to make it less than 00°. If an even multiple of 0.0° is subtracted the name of the original function is retained, but if an odd mul- tiple is subtracted the original function is re- placed by the co-named function. E.g. to ex- GRAPH OF TAN Q^ From the definition of the trigonometric func- tions, it is evident that they bear certain rela- tions one to another. Some of the fundamental ones are, sin'fl -|- cos=9 = I, sin 9 escS = 1, cos 8 sec 9 = 1, tan 9 cot 9= 1, tan = sin 9/cos 9, 1 + tan= 9 = sec- 9, 1 -)- cot- 9 = csc-9, from which many others readily follow. Besides
