7. If tan 0 = tan3, and 3 cos2 = m2 - 1, then 2 8. 8, Given sin 0 = m sin, and tan 0 = n tan o, find sin and cos p. 9. Given sin (0 + a) = m sin 0, find 0 in terms adapted to logarithmic computation. tan & tan ↓ + tan ↓ tan 0 + tan 0 tan 8 = 1, = and 4 cos e cos cos y sin 20+ sin 2 + sin 24. 12. The tangents of two angles are and tively, find the tangent of their sum. respec 15. Express cos 50 in terms of cos 0 and its powers. 16. Express tan 30 in terms of tan 0. 17. Determine 0 from the equation sina + sin (0 − a) + sin (20 + a) - – a) + sin(20+ a) = sin (0+ a) + sin (20 - a). +4), then tan 0, 20. If cos (0) cos = cos (0 tan, tan sin + sin (360 − −) + sin (72o +0) = sin (36o + 0) + sin (72o − 0). 10. cos 9o. Find the numerical values of sin 12o, cos 22o, and sin 2 A + sin 2 B + sin 2 C = 4sin A sin B sin C. 1. (a + b) cos C + (a + c) cos B + (b + c) cos A = a + b + c. 2. If R be the radius of the circle circumscribed about a given triangle ABC, and r the radius of the inscribed circle, then 3. In a plane triangle ABC, given the sum of the sides AC, BC; the perpendicular from the vertex C upon the base AB; and the difference of the segments of the base made by the perpendicular; find the sides of the triangle. Given the vertical angle, the perpendicular let fall from the vertical angle on the base, and the rectangle under the segments of the base; find the remaining angles. 5. If r be the radius of the circle inscribed in a triangle, and r、 r ̧ r、 the radii of three other circles which touch the sides produced, then 6. In a right-angled triangle, the lines drawn from the acute angles to the point of bisection of the opposite sides are a, ẞ, respectively; find the angles. 7*. In the triangle ABC, BC = 236 feet, angle ABC = 26°30′, and angle ACB = 47°15′, find the remaining sides. 8. In the triangle ABC, = AC 5780 feet, AB 7639 feet, and angle ABC = Find the remaining side and angles. 9. 43° 8'. The angles of a triangle are as the numbers 3, 4, 5, and the radius of the inscribed circle is known. Find the area of the triangle. 10. A triangular field ABC, the sides of which are given, is to be divided into two parts in the ratio of 2 : 1, by a fence passing across from a given point D in AC to BC. Find its length, 11. Find the area of a triangle, having given two angles and a side opposite to one of them. 12. Given the distances from the angles, of the point at which the sides of a plane triangle subtend equal angles; find the sides and the area. 13. The sides of a triangle are in arithmetical progression, and the distance of the centres of the inscribed and circumscribed circles is a mean proportional between the greatest and the least; shew that the sides are in the proportion of √5-1:√5:√5 +1. 14. In a right-angled triangle, if the hypothenuse be divided into two segments x and y, by the line which bisects the right angle, and t = the tangent of half the difference of the acute angles, then 15. Given the angles of a triangle, and the radius of the inscribed circle, determine the sides. 16. The triangle ABC has its angles A, B, C in the proportion of 2 : 3 : 4. Prove that cos 17. A a + c 2 2b The angles of a triangle are as the numbers 1, 2, 3; and the perpendicular from the greatest angle on the opposite side is p. Find its area. 18. Determine the triangle, whose sides are three consecutive terms in the series of natural numbers, and whose largest angle is double of the least. 19. If a, b, c be the sides of a triangle, p, q, r perpendiculars from a point within the triangle bisecting the sides, prove that α b с abc 4 p q r 20*. If lines be drawn from the angular points of a triangle to the middle points of the opposite sides, the triangle will be divided into six equal parts. 21. If a perpendicular be let fall from the vertex of any triangle on the base, the rectangle under the sides of |