21. If

(1+cos a) (1+cos B) (1+cos y) = (1 − cos a) (1 − cos B) (1 — cos y), shew that each expression is equal to ±sin a sin ß sin y.

22. If the sum of four angles is 180°, shew that the sum of the products of their sines taken two together is equal to the sum of the products of their cosines taken two together.

*23. In a triangle, shew that

(1) II. II. II3=16R2r;

(2) II2+1,12=16R2.

24. Find the angles of a triangle whose sides are proportional

sin 20.

(2) sin 24, sin 2B,

25. Prove that the expression

sin2 (+a)+sin2 (8+B)-2 cos (a-B) sin (0+a) sin (0+8)

is independent of 0.

*26. If a, b, c, d are the sides of a quadrilateral described about a circle, prove that

27. Tangents parallel to the three sides are drawn to the in-circle. If p, q, r be the lengths of the parts of the tangents within the triangle, prove that 2+

α

+ =1.

с

[The Tables will be required for Examples 28 and 29.]

28. From the top of a cliff 1566 ft. in height a train, which is travelling at a uniform speed in a straight line to a tunnel immediately below the observer, is seen to pass two consecutive stations at an interval of 3 minutes. The angles of depression of the two stations are 13° 14′ 12′′ and 56° 24′ 36′′ respectively; how fast is the train travelling?

29. A harbour lies in a direction 46° 8' 8'6" South of West from a fort, and at a distance of 27.23 miles from it. A ship sets out from the harbour at noon and sails due East at 10 miles an hour; when will the ship be 20 miles from the fort?

π and all angles coterminal with these will have the

6'

same sine.

This example shews that there are an infinite number of angles whose sine is equal to a given quantity. Similar remarks apply to the other functions.

We proceed to shew how to express by a single formula all angles which have a given sine, cosine, or tangent.

237. From the results proved in Chap. IX., it is easily seen that in going once through the four quadrants, there are two and only two positions of the boundary line which give angles with the same sine, cosine, or tangent.

Thus if sin a has a given value, the positions of the radius vector are OP and OP' bounding the angles α and π-a. [Art. 92.]

If cos a has a given value, the positions of the radius vector are OP and OP' bounding the angles a and 2π-a. [Art. 105.]

P'

Р

If tan a has a given value, the positions of the radius vector are OP and OP' bounding the angles a and +a. [Art. 97.]

a

sine.

238. To find a formula for all the angles which have a given

Let a be the smallest positive angle which has a given sine. Draw OP and OP' bounding the angles a and -a; then the required angles are those coterminal with OP and OP'.

The positive angles are

2р+а and 2рñ+(ñ−α),

where p is zero, or any positive integer.

The negative angles are

-(a) and -(2-a),

and those which may be obtained from them by the addition of any negative multiple of 2; that is, angles denoted by

2qT-(+) and 2qπ-(2-a),

where q is zero, or any negative integer.

These angles may be grouped as follows:

and it will be noticed that even multiples of are followed by +a, and odd multiples of π by -α.

Thus all angles equi-sinal with a are included in the formula

where n is zero, or any integer positive or negative.

This is also the formula for all angles which have the same cosecant as a.

Example 1. Write down the general solution of sin @='

√3

The least value of which satisfies the equation is

π

; therefore

the general solution is n + ( − 1)"Ţ.

Example 2. Find the general solution of sin20= sin2a.

This equation gives either sin 0=+sin a...

or sin = sin a=sin(-a)

..(1),

..(2).

Both values are included in the formula 0=nra.

239. To find a formula for all the angles which have a given

cosine.

-a and - (2π − a),

τα

and those which may be obtained from them by the addition of any negative multiple of 2; that is, angles denoted by

2qπ-а and 2qñ — (2π — α),

where q is zero, or any negative integer.

The angles may be grouped as follows :

1(2q-2)+α,

and it will be noticed that the multiples of π are always even, but may be followed by +a or by a.

Thus all angles equi-cosinal with a are included in the formula

where n is zero, or any integer positive or negative.

This is also the formula for all angles which have the same secant as a.

Example 1. Find the general solution of cos 0=

240. To find a formula for all the angles which have a given tangent.

Let a be the smallest positive angle which has a given tangent. Draw OP and OP' bounding the angles a and T+a; then the required angles are those coterminal with OP and OP'.

The positive angles are

2р+а and 2рπ+(π+α),

where Ρ is zero, or any positive integer.

The negative angles are

τα

and those which may be obtained from them by the addition of any negative multiple of 2π; that is, angles denoted by

2qπ-(-a) and 2qπ- (2π-a),

where q is zero, or any negative integer.

The angles may be grouped as follows:

and it will be noticed that whether the multiple of π is even or odd, it is always followed by +a. Thus all angles equi-tangential with a are included in the formula

This is also the formula for all the angles which have the same cotangent as a.

Example. Solve the equation cot 40=cot 0.

The general solution is

whence

40=nπ+0;

241. All angles which are both equi-sinal and equi-cosinal with a are included in the formula 2n+a.

All angles equi-cosinal with a are included in the formula 2na; so that the multiple of π is even. But in the formula n+(-1) a, which includes all angles equi-sinal with a, when the multiple of T is even, a must be preceded by the + sign. Thus the formula is 2nπ +α.