and are equal, since PM P'M' QN example, the three ratios the three triangles are similar, and therefore the sine of the angle AOB has the same value from whichever of the three triangles it is defined. Hence the numerical value of the sine of an angle can depend only on the magnitude of the angle; and the same is true for the other trigonometrical ratios. It is, of course, immaterial, as far as regards the values of the trigonometrical ratios of an angle, whether the angle is expressed in degree measure or in radian measure. For example, 60° and represent the same angle, though in different units, and therefore sin60°=sing, cos60° = cos π π π cos, tan60° The values of the trigonometrical ratios of angles are calculated by the higher parts of Trigonometry, but in a few particular cases the values may be found by simple geometrical constructions. In Arts. 17 and 18 the values of the trigonometrical ratios are found for the angles 60°, 45°, and 30°. The trigonometrical ratios of an angle are also called the trigonometrical functions of the angle. Ex. ABC is a triangle in which the angle BCA is a right angle. The lengths of the two sides AB and BC are 13 feet and 12 feet respectively. Find the values of the trigonometrical ratios of the angle BAC. Comparing this triangle ABC with the triangle OPM of Art. 13, we see that in applying A the definitions to find the trigonometrical 6 5 B 12 ratios of the angle BAC, the side BC corresponds to PM the perpendicular, the side AC corresponds to OM the base, and the side AB corresponds to OP the hypotenuse. We are given the lengths of AB and BC; if we knew also the length of AC, we could write down the values of the trigonometrical ratios of the angle BAC. Now the length of AC can be found by Euclid I. 47. For by that proposition, since the angle ACB is a right angle, sq. on AB=sq. on AC+ sq. on BC; hence if x denote the length of AC in feet, therefore 132 = x2+122; x2 132-122=169-144 = 25; therefore, taking the square root, x or AC=5 feet. We can now find the values of the trigonometrical ratios of the angle BAC, as we know the lengths of the three sides of the right-angled triangle ABC. 1. ABC is a triangle in which the angle BCA is a right angle and the lengths of the two sides AB and BC are 65 inches and 63 inches respectively; find the values of all the trigonometrical ratios of the angle BAC. 2. ABC is a triangle in which the angle BCA is a right angle and the lengths of the two sides BC and CA are 7 feet and 24 feet respectively; find the values of all the trigonometrical ratios of the angle ABC. 3. ABC is a triangle in which the angle BCA is a right angle and the lengths of the two sides BC and CA are a inches and b inches respectively; find the tangent, the sine, and the cosine of the angle BAC. 15. Complement of an Angle. The complement of an angle is the difference between the angle and a right angle. If A is the degree measure of an angle, 90° – A is the degree measure of the complement of the angle; and if 0 is the radian measure of an angle, the radian measure of the complement of the angle. For example, the complements of the four angles 10°, 30°, 40°, 60°, are 80°, 60°, 50°, 30°, respectively. The same four angles π п 2п π expressed in radian measure are and the com 18' 6' 9' 3' plements expressed in radian measure are spectively. 4π п 5п π 9 3' 18' 6' re Two angles are said to be complementary when they are together equal to a right angle. Thus the two acute angles of any right-angled triangle are complementary. 16. Significance of prefix Co- in Cosine, Cotangent, Cosecant. Let ABC be a triangle in which the angle C is a right angle, so that the two angles BAC and ABC are complementary. If A and B denote these two angles, we have by the definitions 7 Hence we see that cosA = sinB, and sinA = cos B, or, the cosine of an angle is equal to the sine of the complement of the angle, and the sine of an angle is equal to the cosine of the complement of the angle. The same relation holds, as the student can easily prove from the figure, for tangent and cotangent, and also for secant and cosecant. Thus the cosine, the cotangent, and the cosecant of an angle are equal respectively to the sine, the tangent, and the secant of the complement of the angle, and these relations are sometimes given as the definitions of the three ratios, cosine, cotangent, and cosecant. If A denotes the degree measure of an angle, 90° – A denotes the degree measure of the complement, and therefore We may also write these results in radian measure. If denote the radian measure of an angle, - denotes the radian π 2 Let ABC be a triangle in which the angle C is a right angle and the sides AC and BC are equal. Then the angles BAC and ABC are each equal to 45° or radians. Let a de 45° A a√2 8 π 4 therefore, taking the square root, AB=a √√2. We now know the lengths of all the sides of this rightangled triangle ABC and we can write down the values of the trigonometrical ratios of the angle BAC or 45° or π 4 The student will notice that the values of the cosine, cotangent, and cosecant are respectively equal to the values of the sine, the tangent, and the secant. This might have been foreseen, as 45° is the complement of 45°. It must also be observed that we get the ratios of 45° as numbers, and therefore the values of these ratios are altogether independent of the value of a. This is in accordance with Art. 14. 18. Ratios for 60° and for 30°. B 9 Let ABC be an equilateral triangle, and from any angular point B let BD be drawn perpendicular to the opposite side AC; then the triangle ABD is a right-angled triangle in which the angle BAD is 60° and the angle ABD is 30°. Let a denote the length of the side AD, which is half of AC or AB, then the length of AB is denoted by 2a, A and the length of the third side aV3 60° a D BD may be found by Euclid I. 47. If x denote the length therefore, taking the square root, a or BD = a √3. The values of the ratios for an angle of 30° may also be found from the same triangle ABD, or they may be written down from the values of the ratios for 60°, since 30° is the complement of 60°. Adopting the latter method, |