V. MATHEMATICAL TRIPOS.

'The three days.' Jan. 1881.

1. Explain, and state the several advantages of, the chief systems of angular measurement in use.

Prove that the circumferences of circles vary as their radii; and mention the approximations to their constant ratio which are practically employed.

Shew that there are eleven pairs of regular polygons which satisfy the condition that the measure of an angle of one in degrees is equal to the measure of an angle of the other in grades: and find the number of sides in each.

2. Define the sine of an angle; and find the value of the sines of angles of 135°, 240°, 2921o, 432o.

Shew that

sin2100+ cos220° - sin 10° cos 20° = sin2 10° + cos2 40° + sin 10° cos 40°=.

3. Prove geometrically that

sin x + sin y=2 sin (x + y) cos (x − y).

Solve the equation

cos x + sin 3x + cos 5x + sin 7x+ .+ sin (4n − 1) x = (sec x + cosec x).

...

4. Find an expression for cos (x1+x+x) in terms of sines and cosines of x1, X2, X3.

State the corresponding theorem for the case of n angles x1, x2, .......¤ ̧· If cos (y-z) + cos (z −x) + cos(x − y) = -, shew that

cos3 (x+2)+cos3 (y+0) + cos3 (z+0) − 3 cos (x + 0) cos (y+0) cos (z+0) vanishes whatever be the value of 0.

5. Shew how to solve a triangle having given the three sides: proving from the formulæ obtained that there cannot be more than one triangle, though there may be none, with the given parts.

The perpendiculars from the angular points of an acute-angled triangle ABC on the opposite sides meet in P: and PA, PB, PC are taken for the sides of a new triangle. Find the condition that this should be possible: and if it is, and the angles of the new triangle are a, B, y, shew that

6. Find the radii of the inscribed, the circumscribed, and the nine-point circles of a given triangle.

If O be the centre of the first, O' of the second, and P the centre of perpendiculars, shew that the area of the triangle OO'P is

– 2R2 sin } ( B – C) sin § (C – A) sin § (A – B),

where R is the radius of the circle circumscribing ABC.

VI. MATHEMATICAL TRIPOS, PART I. June, 1882.

1. Explain the different methods of measuring angles.

Find the number of degrees in each angle of a regular polygon of n sides (1) when it is convex, (2) when its periphery surrounds the inscribed circle m times.

Find correct to 01 of an inch the length of the periphery of a. decagon which surrounds an inscribed circle of a foot radius three times.

2. Prove geometrically the formula

Prove that

cos a + cos ẞ=2 cos § (a+ẞ) cos § (a – ß).

2 cos (a-B) cos (0+ a) cos (0 +ß) +2 cos (B − y) cos (0+ẞ) cos (0+y) +cos (y− a) cos (0+y) cos (0+a) – cos2 (0+a) – cos 2 (0 + ß)

1

- cos2 (0+ y)-1 is independent of 0, and exhibit its value as a product of cosines.

3. Prove geometrically the formula

Prove that if a, ß, y, d be four solutions of the equation

tan (0+1)=3 tan 30,

no two of which have equal tangents, then

and

tana+tan ẞ+tan y+tan 8=0,

tan 2a +tan 28 +tan 2y+tan 28=.

4. Prove that in general the change in the cosine of an angle is approximately proportional to the change in the angle.

Prove that if in measuring the three sides of a triangle small errors x, y be made in two of them a, b, then the error in the angle C will be

-(cot B + /cot 4),

and find the errors in the other angles.

5. Prove that in any triangle a cos B+b cos A=c, and deduce the formula c2=a2+b2-2ab cos C.

Prove that if O be the centre of the circumscribing circle of the triangle ABC, the sides of the triangle formed by the centres of the three circles BOC, COA, AOB will be proportional to

sin 24 sin 2B sin 2C.

Find the angles of the new triangle correct to one second when the sides of the triangle ABC are in the ratio 4: 5 : 7.

6. Find the radius of the inscribed circle of a triangle in terms of one side and the angles.

Prove that if P be a point from which tangents to the three escribed circles of a triangle ABC are equal, the distance of P from the side BC will be

(b+c) sec 4 sin B sin C.

VII. OXFORD AND CAMBRIDGE SCHOOLS EXAMINATION. Eton, 1882.

1. Prove that the cosine and sine of an angle have their signs changed, but their magnitudes unaltered, if the angle be increased by two right angles. Investigate a general formula for all angles whose tangent is equal to tan A.

2. Find the cosine and tangent of 45° and 60o. Apply Euclid v1. 3 to find tan 15o.

3. Prove that sin (A – B) = sin A cos B—cos A sin B, and that

4. Express cos (B+y-a) cos (y+a - ẞ) cos(a+ẞ-y), as the sum of cosines of separate angles.

5. Express sin A in terms of sin A; and prove, à priori, that to any given value of sin A, four values of sin 14 must correspond.

Having given sin 18°=} (√√5−1), find cos 81o.

6. Shew how to find the height of an inaccessible object by observations of its angles of elevation, taken at two points on a straight line through its base.

I stand on a hill on one side of a lake, and observe the angle of elevation (a) of the summit of a mountain across the lake, and also its angle of depression (B) as seen by reflection in the lake. If h be the known height of the mountain, shew that its distance is

it being given that the ray of light from the top of the mountain makes the same angle with the vertical after reflection from the lake as it did before reflection.

7. Express the sine of half an angle of a triangle in terms of the sides.

Prove that, in any triangle,

2 (cos 4-sin 14)2 cos B cos C

=(cos C+cos A – cos B) (cos 4+cos B-cos C).

8. Find the radius of a circle described about a triangle.

If the radius of this circle be equal to the least side of the triangle, what is the magnitude of the least angle?

VIII. OXFORD AND CAMBRIDGE SCHOOLS EXAMINATION. Eton, 1883.

1. Given =3·1416, find the number of degrees in the unit of circular measure of angles.

α

3. Prove that all angles included in the formula 2nπ±a have the same cosine as a.

Solve the equation cos + √3 sin 0=2,

5. ABC is a triangle right-angled at A; BD meets AC in D: find AD in terms of CD and the angles ABC, ABD,

(3) (b-c) cot A+ (c− a) cot B+ (a - b) cot &C=0.

7. Find an expression for the radius of the circumscribed circle of any triangle in terms of the sides.

The bisector of the angle A meets the side BC in D and the cira2 sec A cumscribed circle in E: shew that DE=

2 (b+c) *

8. If the ratio of two sides of a triangle is 2+√3: and the included angle is 60°, find the other angles.

n being a positive integer. What are the other values?