I.

1. Being given of a triangle the base and difference of base angles, and that the sum of its sides is 180°, construct it.

2. Prove the following formula for an equilateral triangle :—

3. Find cos x, in terms of a and b, from the equations

4. Given the base, sum of sides, and one base angle; find the other parts of the triangle.

(Science and Art Hon.)

5. If an angle of a triangle be equal to (or supplemental to) the opposite side, show that

1- sec2 a sec2 b- sec2 c + 2 sec a sec b sec c = 0.

6. Find the locus of the intersection of equal tangents to two small circles.

7. If each leg of a tripod be inclined at an angle to each of the other legs; find the angle between a leg and the horizontal plane, the legs being supposed equal to each other.

3. If A denote one of the angles of an equilateral triangle, and A' an angle of its polar triangle, show that

5. If four points A, B, C, D on a great circle be joined to a point P on the sphere, show that

6. Using the notation of Ex. 5, show that if cot AB, cot AC, cot AD are in Arithmetical Progression, cot APB, cot APC, cot APD are also in Arithmetical Progression, and hence a harmonic row always subtends a harmonic pencil at any point on the sphere.

7. The sum of the sides of a spherical polygon is less than 2π.

8. Prove the following formula for the reduction of the parts of a spherical triangle:

(sec a sin b cos A − sin c)2 + (sec a cos b · cos c)2 (1 cosec2 a sin2 A) tan2 a cos2 B cos2 C.

9. Given

(Educational Times, vol. xxxvii. p. 69.)

a = 72° 44′, b = 36° 22′, C = 56° 12';

calculate A, B and c.

10. Given, in a right-angled triangle, a = 86° 45', b = 108° 20'; calculate A, B, and c.

III.

1. If two small circles on a sphere touch each other, show that the great circle joining their poles passes through their point of contact.

3. Show that the arcs joining the middle points of the sides of the colunar triangles of a given spherical triangle are equal to the corresponding angles of the chordal triangle.

4. Show that a triangle can be constructed having sides 90° a, 90° - B, 90°, and angles 90-a, 90 — 1 b, 90 — 1c, where a, B, y are the arcs joining the middle points of the sides of a triangle whose sides are a, b, c.

5. Find the locus of the middle points of the sides of a triangle, having given the base and the sum of the three angles.

6. Show that

Σ sin a cot XY = 0,

where X is the intercept made by the bisectors of the angle A on the side a of a spherical triangle.

7. Deduce the analogue in plano of Ex. 6.

8. Having given the sides of a triangle, each = 60°; find the segments of the bisectors of the angles.

9. In a right-angled triangle, being given

A = 47° 45', B = 63° 30';

calculate the sides.

10. Given

a = 75° 12', b = 63° 27', c = 52° 23';

calculate A, B, and C.

where ẞ is the bisector of the side 2c drawn from the opposite vertex, im the angle it makes with that side.

(Dublin Univ. Exam. Fajers.,

2. Show that the sum of the three arcs joining the middle points of the sides of the three colunar triangles of a given triangle is equal to two right angles.

4. Given the hypotenuse of a right-angled triangle; prove that the difference of the sides is a maximum when their sum is equal to a quadrant.

5. Find the relation connecting the angles of a triangle if

1.+ cos a + cos b + cos c = 0.

6. Prove that

cos (BC)

=

tana - tan cos C-tanc cos B

cos (AC) tan b-tana cos C - tanc cos A ̊

7. The great circle through the poles of two small circles is perpendicular to the great circle through their points of intersection.

8. If a and a' are the segments of the bisector of the side a of a triangle, show that

V.

1. If a, b, c, d be the sides of a spherical quadrilateral, ♪ and d' its diagonals, and if a and c, b and d, 8 and intersect at angles 0, 0, 4, respectively, show that

sin a sin c cos + sin b sin d cos + sin d sin d' cos = 0.

2. Find the locus of a point P, whose distances from three fixed points A, B, C are connected by the relation

cos AP+cos BP + cos CP = const.

3. If a, a'; B, B'; y, y', be the segments of the perpendiculars to the sides of a spherical triangle drawn from the opposite vertices, show that

=

tan a tan a': = tan ẞ tan B' tan y tan y'.

4. Using the notation of Ex. 3, show that

6. If the bisectors of the angles of a triangle meet at 0, show that the angle AOB is supplemental to the angle COD, where OD is a perpendicular from O on BC.

7. Using the notation of Ex. 6, show that

(1) tan A0 cos BOC = tan BO cos COA = tan CO cos AOB;

(2) sin BOC: sin COA: sin AOB : = cos A: cos B: cos C.

8. The angle in a semicircle is a right angle; what is the analogous theorem on the sphere?

a = 85° 16', b = 63° 24', C = 76° 12';

9. Given

calculate A, B, and c.

10. In a right-angled triangle, given с = calculate a, B, and b.

48° 45', and A

21° 27';