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Whence b ta:a:: dtcic And inversely a : b + a

c: d c....

...(2)
But a : a = b ::cic Id
... a : a

6

C:cu d.

.

66

3. Find the number of different combinations that may be made of n different things taking r of them together.”

Of n different objects, a, b, c, d, &c., a may be placed before each of the rest and thus form n- -1 permutations in which a stands first; there may similarly be ne -1 permutations in which b stands first; and the same fof each of the n different objects ; so that there are n (n-1) permutations of objects taken two and two. (1) If from the n things the first be taken away there will remain nl things ; and the number of permutations that can be formed of them, taken two and two, will he (n-1) (1–2). Now let the object a which we have assumed to be taken from the n things be prefixed to each of the permutations (n-1) (n-2), and there will be (n-1) (142) such permutations taken three together, in which a stands first; there will be as many in which b stands first, and so on for the rest; so that, on the whole,

n (11-1) (n—2) = number of permutations of n different things taken three together.

We thus perceive that the number of permutations o n things taken two together

-1)
three together ..n(n-1) (n—2)

n

n

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.. n

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*

.

Similarly it is shown,
four together ..n -1)

-3)
five together -1) (n–2)(n- -3) (144)

4 r together = n(n-1) (1-2)(n-3) (n-4) (1-5) .(n—r—1)

=n(n-1) (n-2)(n-3)....(n-r + 1) For each combination of things taken two and two there are twice as many permutations; for each combination, as a b, admits of two permutations b a, a b. Hence the permutations of things taken two and two must be divided by two to give the combinations of the same objects; or the number of combinations of n

n (n-1) things taken two and two together

2 There are n (n-1) (n—2) permutations of n things taken three and three together, and each combination of three admits 3 X 2 X 1 permutations, so that the permutations must be divided by 3 X 2 X 1 to give the combinations; that is, the number of combinations of n things taken three and three, is

n (n-1) (1-2)

1 2 3

n (n-1) (n-2)(n-3) taken four together =

1.2.3.4. And similarly the number of combinations of n things taken r times together n (n-1) (n—2)(n-3) (n-p+1)

1 2.3.4 Example ; required the number of parties of three that may be formed out of six persons. Here n = 6 and r = 3

n (1-1) (n-2) 6 X 5 X 4 So that

= 20. 1.2.3.

1.2

SECTION III. 1. “In a circle the angle in a semicircle is a right

r

3

angle, but the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle."

See Euclid iii. 31.
2. “Describe a circle in a given triangle.”

See Euclid iv. 4. Tate's Geometry and Mensuration, Art. 69.

3. “Similar triangles are to each other in the duplicate ratio of their homologous (or corresponding) sides. Prove this, and indicate the steps of the proof of the corresponding proportion for all similar figures."

See Euclid vi. 19. Tate, Art. 55. Geometry.

The first step in the proof of the proposition that similar figures are to one another in the duplicate ratio of their corresponding sides, is to divide the similar figures into similar triangles and show that they are similar; secondly, the corresponding triangles of each polygon being proved similar and consequently in the duplicate ratio of their homologous sides to show that collectively, the triangles are in the same ratio that they were proved to be in severally. But the triangles collectively are the polygons individually; hence similar polygons are to one another as the squares of their like sides.

SECTION IV. 1. (a) “ Define the sine and the tangent of an angle.

(6) “What are the values of sin. 90°, tan. 180°, and tan, 45° ? (c) “Prove that cos, 2 A = 2 cos. 2A —1:

=1

2 sin. 2A."

(a) The sine of an arc (which is the measure of an angle) is the perpendicular let fall from one extremity of that arc upon the radius passing through the other extremity. The tangent of an arc (or of an angle) is the straight line which touches the circle at one end of the arc, at right angles with the radius that passes through the point where the tangent touches the arc, and is terminated by the radius produced through the other end of the arc.

(6) Sin. 90o = radius or 1, according to the standard of trigonometrical measurement.

Tan. 180° 0; tan. 45o = radius or 1. (c) The reply to this part of the question requires, as a preliminary step, either to prove or to assume one of the fundamental formulæ of analytical trigonometry.

For the sake of conciseness, and to dispense with the necessity of a figure, we adopt the latter course.

The formula from which the expression in the question

may be most directly deduced is, Cos. (A + B) =

= cos. A. cos. B. sin. A. sin. B. In this formula, suppose

B = A; then cos. (A + A) = cos. A. cos. A — sin. A. sin. A.

... Cos. 2 A = cos.? A - sin.? A. (1) In any right-angled triangle the sides which contain the right angle are respectively the sine and cosine of one of the acute angles, the hypotenuse being radius. In trigonometrical calculations radius is taken unity. Hence by Euclid i. 47. Sin.' + cos.' = 1, and

Sin.? =1—cos.? Adopting this value of sin.” A in formula (1) we get, Cos. 2 A = cos. A-(1 — cos. A)= cos. A – 1

+ cos. 2 A = 2 cos. A - 1. .... (2) Also, cos. = 1 - sin.' ; substituting this in (2)

Cos. 2 A = 1 — 2 sin. A .. cos. 2 A = 2 cos. 2A 1=1- 2 sin. ? A.

2. “Having given one side of a right-angled triangle and the angle adjacent to that side, show by what calculations the other parts of the triangle may be found.”

1st. Since the three angles of a triangle equal 180, and the given triangle is right angled, the remaining two angles = 90°; so that one of them being given

2

the other is obtained by subtracting the known one from 90°.

2nd. If the given side be the hypotenuse, multiply it by the sine of the adjacent angle, and the product will be the side opposite the angle whose sine is used : multiply it by the cosine of the same angle and the product will be the side adjacent to the angle.

3rd. If one of the other sides is given, divide it by the sine of the adjacent angle, the quotient will be the opposite side: divide it by the cosine of the same angle, the quotient will be the adjacent side. These results are evident from the definition of the sine, cosine, and tangent of an angle. See Hymer's Trigonometry, page 53. Hall's and Snowball's are most valuable text-books. Tate's compendious summary of trigonometry, in his Geometry and Mensuration, will be found of great utility as a first course in this department of mathematical science.

3. (a) “Write down the expression for the cosine of an angle of a triangle in terms of the sides and prove the expression for the sine of an angle and for the area of a triangle, in terms of the sides.

(6) “What is the logartihm of a number? What are the properties of logarithms on which the utility of logarithmic tables depends ?”

If A, B, C, be the angles of a triangle, and a, b, c, the sides of the same; then,

62 + c a? Cos. A =

2 b c

a2 + c2 62 Cos. B. =

2 ac

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In the triangles A B C, let fall the perpendiculars CD,

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