89. To trace the changes in sign and magnitude of tan A as A increases from 0° to 360°. With the figure of Art. 86, tan A = MP OM' therefore depend on those of MP and OM. and its changes will When A=0°, MP=0, 0M=r; ... tan 0° =-= 0. In the first quadrant, MP is positive and increasing, OM is positive and decreasing; .. tan A is positive and increasing. When A=90°, MP=r, 0M=0; .. tan 90°: In the second quadrant, MP is positive and decreasing, OM is negative and increasing; .. tan A is negative and decreasing. When A = 180°, MP=0; .. tan 180°=0. MP is negative and increasing, OM is negative and decreasing; .. tan A is positive and increasing. When A=270°, OM=0; .. tan 270° =∞. In the fourth quadrant, MP is negative and decreasing, OM is positive and increasing; .. tan A is negative and decreasing. When A=360°, MP=0; ... tan 360° 0. NOTE. When the numerator of a fraction changes continually from a small positive to a small negative quantity the fraction changes sign by passing through the value 0. When the denominator changes continually from a small positive to a small negative quantity the fraction changes sign by passing through the value ∞. For instance, as A passes through the value 90°, OM changes from a OM small positive to a small negative quantity, hence that is cos A, OP changes sign by passing through the value 0, while tan A, changes sign by passing through the value ∞. 90. The results of Art. 89 are shewn in the following diagram : The student will now have no difficulty in tracing the variations in sign and magnitude of the other functions. 91. In Arts. 86 and 89 we have seen that the variations of the trigonometrical functions of the angle XOP depend on the position of P as P moves round the circumference of the circle. On this account the trigonometrical functions of an angle are called circular functions. This name is one that we shall use frequently. EXAMPLES. IX. Trace the changes in sign and magnitude of 1. cot A, between 0° and 360°. 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 OA Ot OB -=tan POA; that is, ATOA × tan POA; =sec BOP=cosec POA; that is, Ot=OB x cosec POA. 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. MISCELLANEOUS EXAMPLES. C. 1. 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' 1800 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 OP'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; |