=

. 6. In a triangle ABC, given a = = 309, b = 450, and A 27° 50', to first determine whether the triangle admits of more than one solution, and then solve it.

June, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. Write the values of the different trigonometric functions for angles of 0°, 90°, 180°, 270°, 300°.

2. Write the simplest equivalent expressions for

sin

(——a), cos (5+a), tan (-a), sin(2π-α).

3. Write the fundamental formulæ connecting two or more of the trigonometric functions of the same angle, and deduce any two of them.

5. Deduce the formulæ,

cos 2a = 1 — 2 sin2a; sina=√(1— cos a).

7. In a plane triangle the side a is 4575, the side b is 5656.7, and the angle B (opposite b) is 125° 25'; find the remaining parts of the triangle.

September, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. Find the angle 1.8 in degrees.

2. Calculate the different functions of

and 꿍ᅲ. 6

3. Represent the different trigonometric functions of an

angle in the second quadrant by lines.

6. In a triangle ABC, given the sides a, b, and c, respectively, opposite the angles A, B, and C, as follows:

a =766.3, b = 672, c = 565.05, to find the angle A.

June, 1882.

[State what text-book you have studied on the subject, and to what extent.]

1. Define the different trigonometric functions of an angle.

2. Give the algebraic sign of each function of an angle terminating in each of the four quadrants.

3. Show that sin (π + a) :

= COS a.

- sin a; and that cos (2π — a)

-=

4. Find the formula for tan (a +ẞ) in terms of tan a and tan B. Find also the formula for tan 2 a in terms of tan a.

5. Given 2a tan sin 0-2 a cos 0 + b = 0, to find 0.

6. Given two sides of a plane triangle 180 and 100, and the angle opposite the former 127° 33', to find the other parts.

September, 1882.

[State what text-book you have studied, and to what extent.]

1. Find the number of degrees in the angle subtended at the centre of a circle whose radius is 10 ft. by an arc 9 in. long.

2. Determine all the functions of the following angles :

30°, 7, §, 585°.

3. Given the functions of any angle to find the functions of half that angle.

6. In a plane triangle, given a=100, c=200, and B: 112° 30', to find the remaining parts.

June, 1883.

=

1. Explain the circular measure of an angle. Express an angle of 1.5 in degrees.

2. Give the algebraic signs of the trigonometric functions for angles terminating in each of the four quadrants, and the values of the functions of 0,

π

3п

2'

2

3. State some of the fundamental relations existing between the different functions of the same angle.

4. Find an angle whose tangent is double its sine.

5. Write the different formulae which serve to solve rightangled triangles, and explain their application.

6. Find a formula for expressing sin a in terms of cos a. 7. If a, b, and c are the sides of a triangle, and a the angle opposite the side a, show that

a2 = b2 + c2 — 2bc cos a.

Also state any application you have made of this formula.

8. In a triangle given A = 32°, a=40, b = 50, to find one solution of the triangle. Is there more than one? If so, how do you recognize the fact?

September, 1883.

1. Express an angle of 75° in radians.

2. Express each of the functions of of a.

(+a) by a

function

3. Form a general expression for all angles which have the same cosine.

4. What must the value of the cosine be in order that the tangent may be n times the sine?

5. To find a formula for the tangent of the sum of two angles in terms of the tangents of the angles; and from this formula to derive an expression for the tangent of twice an angle in terms of the tangent of the angle.

6. In a triangle ABC, given the angle A = 118° 04', and the adjacent sides, b= 960, and c = 1686, to find the remaining angles and side of the triangle.

LATIN.

July, 1880.

I. Translate as literally as possible:

a. Quum ea ita sint, tamen, si obsides ab iis sibi dentur, uti ea, quae polliceantur, facturos intelligat, et si Aeduis de injuriis, quas ipsis sociisque eorum intulerint, item si Allobrogibus satisfaciant, sese cum iis pacem esse 5 facturum. Divico respondit: Ita Helvetios a majoribus suis institutos esse, uti obsides accipere, non dare, consuerint; ejus rei populum Romanum esse testem.

b. Ibi vadis repertis partem suarum copiarum transducere conati sunt, eo consilio, uti, si possent, castellum, 10 cui praeerat Quintus Titurius legatus, expugnarent, pontemque interscinderent; si minus potuissent, agros Remorum popularentur, qui magno nobis usui ad bellum gerendum erant, commeatuque nostros prohiberent.

c. Impeditis hostibus propter ea, quae ferebant, onera, 15 subito quabus portis eruptionem fieri jubet. Factum est opportunitate loci, hostium inscientia ac defatigatione, virtute militum, et superiorem pugnarum exercitatione, ut ne unum quidem nostrorum impetum ferrent ac statim terga verterent.

20

d. Dum in his locis Caesar navium parandarum causa moratur, ex magna parte Morinorum ad eum legati venerunt, qui se de superioris temporis consilio excusarent, quod homines barbari et nostrae consuetudinis imperiti bellum populo Romano fecissent, seque ea, quae impe25 rasset, facturos pollicerentur.

e. Britanniae pars interior ab iis incolitur, quos natos in insula ipsi memoria proditum dicunt; maritima pars ab iis, qui praedae ac belli inferendi causa ex Belgis transierant; qui omnes fere iis nominibus civitatum ap