Note on the old definitions of the Trigonometrical Functions. Formerly, Mathematicians considered the trigonometrical functions with reference to the arc of a given circle, and did not regard them as ratios but as the lengths of certain straight lines drawn in relation to this arc. N B MA Let OA and OB be two radii of a circle at right angles, and let P be any point on the circumference. Draw PM and PN perpendicular to OA and OB respectively, and let the tangents at A and B meet OP produced in T and t respectively. The lines PM, AT, OT, AM were named respectively the sine, tangent, secant, versed-sine of the arc AP, and PN, Bt, Ot, BN, which are the sine, tangent, secant, versed-sine of the complementary arc BP, were named respectively the cosine, cotangent, cosecant, coversed-sine of the arc AP. As thus defined each trigonometrical function of the arc is equal to the corresponding function of the angle, which it subtends at the centre of the circle, multiplied by the radius. Thus and AT ОА Ot =tan POA; that is, AT=OA × tan POA; =sec BOP=cosec POA; that is, Ot=OB × cosec POA. OB The values of the functions of the arc therefore depended on the length of the radius of the circle as well as on the angle subtended by the arc at the centre of the circle, so that in Tables of the functions it was necessary to state the magnitude of the radius. The names of the trigonometrical functions and the abbreviations for them now in use were introduced by different Mathematicians chiefly towards the end of the sixteenth and during the seventeenth century, but were not generally employed until their re-introduction by Euler. The development of the science of Trigonometry may be considered to date from the publication in 1748 of Euler's Introductio in analysin Infinitorum. The reader will find some interesting information regarding the progress of Trigonometry in Ball's Short History of Mathematics. 1. MISCELLANEOUS EXAMPLES. C. Draw the boundary lines of the angles whose tangent is 3 equal to and find the cosine of these angles. - 4' 2. Shew that cos A (2 sec A + tan A) (sec A − 2 tan A)=2 cos A – 3 tan A. 3. Given C=90°, b=10·5, c=21, solve the triangle. 5. The latitude of Bombay is 19° N.: find its distance from the equator, taking the diameter of the earth to be 7920 miles. 6. From the top of a cliff 200 ft. high, the angles of depression of two boats due east of the observer are 34° 30′ and 18° 40': find their distance apart, given 7. If A lies between 180° and 270°, and 3 tan A=4, find the value of 2 cot A-5 cos A+ sin A. 8. Find, correct to three decimal places, the radius of a circle in which an arc 15 inches long subtends at the centre an angle of 71° 36′ 3′6′′. 10. The angle of elevation of the top of a tower is 68° 11', and a flagstaff 24 ft. high on the summit of the tower subtends an angle of 2° 10' at the observer's eye. Find the height of the tower, given tan 70° 21' 2.8, cot 68° 11'='4. H. K. E. T. 6 CHAPTER X. CIRCULAR FUNCTIONS OF CERTAIN ALLIED ANGLES. 92. Circular Functions of 180° – A. Take any straight line XOX', and let a radius vector starting from OX revolve until it has traced the angle A, taking up the position X'M' ΤΑ M X Again, let the radius vector starting from OX revolve through 180° into the position OX' and then back again through an angle A taking up the final position OP'. Thus XOP' is the angle 180°-A. From P and P' draw PM and P'M' perpendicular to XX'; then by Euc. 1. 26 the triangles OPM and ÕP'M' are geometrically equal. but M'P' is equal to MP in magnitude and is of the same sign; and OM' is equal to OM in magnitude, but is of opposite sign; 93. In the last article, for the sake of simplicity we have supposed the angle A to be less than a right angle, but all the formulæ of this chapter may be shewn to be true for angles of any magnitude. A general proof of one case is given in Art. 102, and the same method may be applied to all the other cases. 94. If the angles are expressed in radian measure, the formulæ of Art. 92 become Example 1. Find the sine and cosine of 120°. 95. DEFINITION. When the sum of two angles is equal to two right angles each is said to be the supplement of the other and the angles are said to be supplementary. Thus if A is any angle its supplement is 180°- A. 96. The results of Art. 92 are so important in a later part of the subject that it is desirable to emphasize them. We therefore repeat them in a verbal form: the sines of supplementary angles are equal in magnitude and are of the same sign; the cosines of supplementary angles are equal in magnitude but are of opposite sign; the tangents of supplementary angles are equal in magnitude but are of opposite sign. 97. Circular Functions of 180° +A. Take any straight line XOX' and let a radius vector starting from OX revolve until it has traced the angle A, taking up the position OP. Again, let the radius vector starting from OX revolve through 180° into the position M' X' P M X OX', and then further through an angle A, taking up the final position OP'. Thus XOP' is the angle 180° +A. From P and P' draw PM and P'M' perpendicular to XX'; then OP and OP' are in the same straight line, and by Euc. I. 26 the triangles OPM and OP'M' are geometrically equal. and M'P' is equal to MP in magnitude but is of opposite sign; and OM' is equal to OM in magnitude but is of opposite sign; Expressed in radian measure, the above formulæ are written sin (+0)=sin 0, COS (+8)= cos 0, tan (+8)=tan 0. In these results we may draw especial attention to the fact that an angle may be increased or diminished by two right angles as often as we please without altering the value of the tangent. Example. Find the value of cot 210°. cot 210° cot (180° +30°) = cot 30°=√3. |