EXAMPLES. XV. Find all the angles between 0 and 360° which satisfy the following twelve equations: 15. If a point be taken within a triangle so that the sides subtend equal angles at it, and a, ß, y be the distances of this point from the angular points of the triangle, shew, that a2+b2+c2 = 2 (a2 +ß2+y2)+· 4 area of the triangle 16. If a triangle be divided into any two parts by a straight line drawn from one of the angles, shew that the radii of the circles described about these two triangles are in a ratio which is constant for all positions of the dividing straight line. XVI. Changes in the Ratios as the angle changes. 150. In the present Chapter we shall trace the changes in magnitude and sign of the various Trigonometrical Ratios as the angle changes from zero to four right angles. 151. To trace the changes in the sine of an angle as the angle varies. Let BAB' and CAC' be two straight lines at right angles, and suppose a straight line AP of constant length to turn round one end A from the fixed position AB, so that P traces out the circle BCB'C'. From any position of P draw PM perpendicular to BAB; then When AP coincides with AB the perpendicular PM vanishes; then when the angle is zero so also is its sine. While AP moves through the first quadrant PM is positive, and continually increases until AP coincides with AC, and then PM is equal to AP; thus as the angle increases from 0 to 90° the sine increases from 0 to 1. While AP moves through the second quadrant PM is positive and continually decreases until AP coincides with AB', and then PM vanishes; thus as the angle increases from 90° to 180° the sine diminishes from 1 to 0. While AP moves through the third quadrant PM is negative and increases numerically until AP coincides with AC; thus as the angle increases from 180° to 270° the sine is negative and increases numerically from 0 to -1. While AP moves through the fourth quadrant PM is negative and decreases numerically until AP coincides with AB; thus as the angle increases from 270° to 360° the sine is negative and decreases numerically from 1 to 0. 152. To trace the changes in the cosine of an angle as the angle varies. AM With the figure of Art. 151 we have cos PAB= AP' At first AP coincides with AB and then AM=AP, thus when the angle is zero the cosine is 1. While AP moves through the first quadrant AM is positive and continually decreases until AP coincides with AC and then AM vanishes; thus as the angle increases from 0 to 90° the cosine diminishes from 1 to 0. While AP moves through the second quadrant AM is negative and increases numerically until AP coincides with AB'; thus as the angle increases from 90° to 180° the cosine is negative and increases numerically from 0 to -1. While AP moves through the third quadrant AM is negative and decreases numerically until AP coincides with AC'; thus as the angle increases from 180° to 270° the cosine is negative and decreases numerically from -1 to 0. While AP moves through the fourth quadrant AM is positive and continually increases until AP coincides with AB; thus as the angle increases from 270° to 360° the cosine is positive and increases from 0 to 1. 153. To trace the changes in the tangent of an angle as the angle varies. With the figure of Art. 151 we have tan PAB= PM = AM' At first AP coincides with AB, and then AM=AB; thus when the angle is zero so also is its tangent. While AP moves through the first quadrant PM and AM are positive; PM continually increases and AM continually decreases until AP coincides with AC; thus as the angle increases from 0 to 90° the tangent increases from 0 without limit, so that by taking an angle sufficiently near to 90° we can make the tangent as great as we please; this is usually expressed for the sake of abbreviation thus: the tangent of 90° is infinite. While AP moves through the second quadrant PM is positive and AM is negative; PM continually decreases and AM increases numerically until AP coincides with AB'; thus as the angle increases from 90° to 180° the tangent is negative and decreases numerically from an indefinitely large value to zero. While AP moves through the third quadrant PM and AM are negative; PM increases numerically and AM decreases numerically until AP coincides with AC'; thus as the angle increases from 180° to 270°, the tangent is positive and increases from 0 without limit, so that by taking an angle sufficiently near to 270° we can make the tangent as great as we please; this as before is abbreviated thus the tangent of 270° is infinite. While AP moves through the fourth quadrant PM is negative and AM is positive; PM continually decreases numerically, and AM increases until AP coincides with AB; thus as the angle increases from 270° to 360° the tangent is negative and decreases numerically from an indefinitely large value to zero. Similarly the changes in the cotangent of an angle may be traced. 154. To trace the changes in the secant of an angle as the angle varies. 1 The changes in the secant of an angle may be traced by means of the figure in the same way as those of the sine, cosine, and tangent; or we may use the formula sec PAB= and infer the changes in the secant from the known changes in the cosine; we will adopt the latter method. As the angle increases from 0 to 90° the COS PAB' cosine diminishes from 1 to 0; thus the secant increases from 1 without limit, so we may say the secant of 90° is infinite. As the angle increases from 90° to 180° the cosine is negative and increases numerically from 0 to −1; thus the secant is negative and decreases numerically from an indefinitely large value to -1. As the angle increases from 180° to 270° the cosine is negative and decreases numerically from 1 to 0; thus the secant is negative and increases numerically from -1 to infinity. As the angle increases from 270° to 360° the cosine is positive and continually increases from 0 to 1; thus the secant is positive and diminishes from infinity to 1. Similarly the changes in the cosecant of an angle may be traced. 155. To trace the changes in the versed sine of an angle as the angle varies. Since vers A=1-cos A, as the angle increases from 0 to 180° the versed sine increases from 0 to 2, and as the angle increases from 180° to 360° the versed sine diminishes from 2 to 0. 156. Thus we see that the sine and the cosine may have any value between 1 and +1; the tangent and the cotangent may have any value between - ∞ and +; the secant and the cosecant may have any value between - and -1 and between +1 and +∞. And it will be found on examination that no Trigonometrical Ratio changes its sign except when it passes through the value zero or the value infinity. The versed sine is always positive and may have any value between 0 and 2. 157. The student should carefully remember the signs of the Trigonometrical Ratios in the four quadrants: the following table exhibits them. |