= 5. Given sin a = and cos B ; find sin (a + B) and cos (a + B). Solution. Construct the right triangles ABC and DEF, Fig. 80, with a an acute angle of ▲ ABC, and ß an acute angle of ▲ DEF. By [13], sin (a + B) sin a cosẞ+ cos a sin B. Substituting the values for sin a, cos a, sin ß, and cos ẞ from the triangles, sin (a +8)= •fs + † • }} = 15 + 1} = 83. 30°. 6. Given sin a = FIG. 80. }, and cos ẞ = √2; find sin (a + B) and cos (a + B). 7. Find the sine and cosine of 75°, having given the functions of 45' and 8. Given cos α = , and cos 8 and cos (a - ẞ). Ans. ¿ (√3+2 √2), † (1 9. Show that sin (45° + 30°) # sin 45° + sin 30°. 10. Show that sin (25° + 37°) # sin 25° + sin 37°. 13. Prove from Fig. 78 that cos (a + B) cos a + cos B. Nole. A very common mistake made by beginners in trigonometry is to assume that sin (a + ẞ) = sin a + sin ß, etc. Exercises 10, 11, 12, and 13 are given for the purpose of impressing the student with the fact that such relations are not true. 90° +30°. In the following exercises, the angles may have any values. = = 120° + 30°, (b) 150° = 210° — 60°, 16. Find sin 240° and cos 240° by using (a) 240° = 210° + 30°, (b) 240° = 300° - 60°, etc. = 17. Given sin α = -, cos B -1, a in the third quadrant, and ẞ in the second. Find sin (a + B), cos (a + B), cos (a Ans. sin (a + B) B), sin (a - ẞ). 18. Given sin α = , a in the second quadrant, third quadrant. Find sin (a + B), cos (a + ß), sin Ans. sin (a + B) = = 1, cos (a + B) and tan ẞ = fz, 8 in the (a — ẞ), and cos (a = Find the value of 0 in the following exercises. 20. cos 50° cos (85° — a) — sin 50° sin (85° — α) = cos 0. Ans. 0 135° - α. 21. sin (90° + B) cos (90° — ¦ B) + cos (90° + § ß) sin (90° — } }) sin 0. Ans. 0 = 180°. 22. cos (45° - x) cos (45° + x) — sin (45° — x) sin (45° + x) = cos 0. 24. By means of [13], [14], [15], and [16] prove the following relations. Find the value of the following expressions, using only the principal values 70. Formulas for the tangents of the sum and the difference of two angles. - By [7], [13], and [14], tan (a + B) = sin (a + B) = sin a cos B+ cos a sin ẞ Dividing both numerator and denominator by cos a cos B, and Since formulas [13], [14], [15], and [16] are true for all values of a and 8, the formulas [17] and [18] are true in general. EXERCISES 1. Find tan 75°, and tan 15° by means of 45° and 30°. find tan (a + B) and tan (a 2 + √3; tan 15° = 2 – √3. , a and ẞ acute angles, find tan (a + B), Ans. tan (a + B) = = -fi, tan (a - ß) = }}. -, a in the fourth and 8 in the third quadrant, B). Ans. tan (a+B) = -, tan (a-B) = ∞. In the following problems use only the values of the angles < 90°. 4. If α = Solution. if possible. Take the function of (a + B) which involves the given functions the tangent of each side of the equation and substitute for tan a and tan ẞ 19 4 = tan-1 (n2 + n + 1). 71. Functions of an angle in terms of functions of half the angle. Since the formulas for the sum of two angles are true for all values of a and ẞ they will be true when ẞ= a. This formula may be stated as follows: The sine of any angle is equal to twice the product of the sine and cosine of the half angle. And conversely, 2 sin 3 a cos 3 a = sin 2 (3 a) = sin 6 a. 2 sin 25° cos 25° = sin 2 (25°) = sin 50°. cos2 a -(1- cos2 a) sin a sin a sin2 a sin2 a = 1. = 2 cos2 a 1. Given the functions of 30°, to find the functions of 60°. 2. Find the functions of 120°, 180°, 240°, 270° and 300° by means of the preceding formulas. 3. Given tan = 1, 0 < 90°; find the sine, cosine, and tangent of 2 0. Ans. sin 20 = 13, cos 20 = 25, tan 20 = 24. Suggestion. Construct a right triangle with @ as one of the acute angles. 4. Prove that 2 tan a = sin 2 a. |