102. Functions of Half an Angle. If we substitute for x in the formulas cos2 x + sin2 x = 1 (§ 14) and cos2 x sin2x = cos 2x (§ 101), so as to find the functions of half an angle, we have cos2+sin2 z=1, These four formulas are important and should be memorized. From the formula for tan can be derived a formula which is occasionally used in dealing with very small angles. In the triangle ACB we have Exercise 45. Functions of Half an Angle Given sin 30°, find the values of the following: 29 1. sin 15°. 2. cos 15°. 3. tan 15°. 4. cot 15°. Given tan 45° = 1, find the values of the following: 5. cot 74°. 6. sin 22.5°. 7. cos 22.5°. 8. tan 22.5°. 9. cot 22.5°. 10. cot114° 11. Given sin x = 0.2, find sinx and cos x. find com only 12. Given cos x = 0.7, find sinx, cos x, tan x, and cotx. 103. Sums and Differences of Functions. Since we have (§§ 92, sin(x + y) = sin x cos y + cos x sin y, Similarly, by using the formulas for cos (x + y), we obtain and - 2 cos x cos y, cos (x + y) + cos(x − y) = By letting x + y = A, and x y = B, we have x = y = † (A — B), whence 97) (A + B), and This is one of the most important formulas in the solution of oblique triangles. If A, B, C are the angles of a triangle, prove that: 5. sin A+ sin B + sin C 6. cos A+ cos B + cos C = 4 cos A cos B cos C. 1+ 4 sin 4 sin B sin C. 7. tan Atan B+ tan C = tan A tan B tan C. 14. In the triangle ABC prove that cot A+cot B+ cot C = cot A cot B cot C. Change to a form involving products instead of sums, and hence more convenient for computation by logarithms : 33. Prove that cos 45° + cos 75° = cos 15o. 34. Prove that 1 + tan x tan 2 x = tan 2 æ cot x − 1. Prove the following formulas: ·y). 35. (cos x + cos y)2 + (sin x + sin y)2 = 2 + 2 cos (x 38. sin(x + y) cos y cos(x + y) sin y = sin x. CHAPTER VII THE OBLIQUE TRIANGLE 104. Geometric Properties of the Triangle. In solving an oblique triangle certain geometric properties are involved in addition to those already mentioned in the preceding chapters, and these should be recalled to mind before undertaking further work with trigonometric functions. These properties are as follows: The angles opposite the equal sides of an isosceles triangle are equal. If two angles of a triangle are equal, the sides opposite the equal angles are equal. If two angles of a triangle are unequal, the greater side is opposite the greater angle. If two sides of a triangle are unequal, the greater angle is opposite the greater side. A triangle is determined, that is, it is completely fixed in form and size, if the following parts are given : 1. Two sides and the included angle. 2. Two angles and the included side. 3. Two angles and the side opposite one of them. 4. Two sides and the angle opposite one of them. 5. Three sides. The fourth case, however, will be recalled as the ambiguous case, since the triangle is not in general completely determined. If we have given ▲A and sides a and b in this figure, either of the triangles ABC and AB'C will satisfy the conditions. If a is equal to the perpendicular from C on AB, however, the points B and B' will coincide, and hence the two triangles become congruent and the triangle is completely determined. B' a C B The five cases relating to the determining of a triangle may be summarized as follows: A triangle is determined when three independent parts are given. This excludes the case of three angles, because they are not independent. That is, A = 180° — (B + C), and therefore A depends upon B and C. 108 PLANE TRIGONOMETRY 105. Law of Sines. In the triangle ABC, using either of the figures as here shown, we have the following relations. Therefore, whether h lies within or without the triangle, we obtain, by division, the following relation: In the same way, by drawing perpendiculars from the vertices A and B to the opposite sides, we may obtain the following relations: This relation between the sides and the sines of the opposite angles is called the Law of Sines and may be expressed as follows: The sides of a triangle are proportional to the sines of the opposite angles. and this is frequently given as the Law of Sines. It is also apparent that a sin B = b sin A, a sin C = c sin A, and b sin C = c sin B, three relations which are still another form of the Law of Sines. |