20. If A denote any acute angle, we have proved that all the trigonometrical ratios of A depend only on the magnitude of the angle A and not upon the lengths of the lines which bound the angle. It may easily be seen that a change made in the value of A will produce a consequent change in the values of all the trigonometrical ratios of A. This point will be discussed more fully in Chap. IX.

DEFINITION. Any expression which involves a variable quantity x, and whose value is dependent on that of x is called a function of x.

Hence the trigonometrical ratios may also be defined as trigonometrical functions; for the present we shall chiefly employ the term ratio, but in a later part of the subject the idea of ratio is gradually lost and the term function becomes more appropriate.

21. The use of the principle proved in Art. 19 is well shewn in the following example, where the trigonometrical ratios are employed as a connecting link between the lines and angles.

Example. ABC is a right-angled triangle of which A is the right angle. BD is drawn perpendicular to BC and meets CA produced in D: if AB=12, AC=16, BC=20, find BD and CD.

The same results can be obtained by the help of Euc. vI. 8.

EXAMPLES. II.

1. The sides AB, BC, CA of a right-angled triangle are 17, 15, 8 respectively; write down the values of sin A, sec A, tan B,

sec B.

2. The sides PQ, QR, RP of a right-angled triangle are 13, 5, 12 respectively: write down the values of cot P, cosec Q, cos Q, cos P.

3. ABC is a triangle in which A is a right angle; if b=15, c=20, find a, sin C, cos B, cot C, sec C.

4. ABC is a triangle in which B is a right angle; if a = 24, b=25, find c, sin C, tan A, cosec A.

5. The sides ED, EF, DF of a right-angled triangle are 35, 37, 12 respectively: write down the values of sec E, sec F, cot E, sin F.

6. The hypotenuse of a right-angled triangle is 15 inches, + and one of the sides is 9 inches: find the third side and the sine, cosine and tangent of the angle opposite to it.

7. Find the hypotenuse AB of a right-angled triangle in which AC-7, BC=24. Write down the sine and cosine of A, and shew that the sum of their squares is equal to 1.

8. A ladder 41 ft. long is placed with its foot at a distance of 9 ft. from the wall of a house and just reaches a window-sill. Find the height of the window-sill, and the sine and cotangent of the angle which the ladder makes with the ground.

9. A ladder is 29 ft. long; how far must its foot be placed from a wall so that the ladder may just reach the top of the wall which is 21 ft. from the ground? Write down all the trigonometrical ratios of the angle between the ladder and the wall.

10. ABCD is a square; C is joined to E, the middle point of AD: find all the trigonometrical ratios of the angle ECD.

11. ABCD is a quadrilateral in which the diagonal AC is at right angles to each of the sides AB, CD: if AB=15, AC=36, AD=85, find sin ABC, sec ACB, cos CDA, cosec DAC.

12. PQRS is a quadrilateral in which the angle PSR is a right angle. If the diagonal PR is at right angles to RQ, and RP 20, RQ=21, RS=16, find sin PRŠ, tan RPS, cos RPQ, cosec PQR.

CHAPTER III.

RELATIONS BETWEEN THE TRIGONOMETRICAL RATIOS.

22.

Reciprocal relations between certain ratios.

(1) Let ABC be a triangle, right-angled at C;

23. To express tan A and cot A in terms of sin A and cos A.

From the adjoining figure we have

=

which is also evident from the reciprocal relation cot A =

Example. Prove that cosec A tan A=sec A.

1

tan A

24. We frequently meet with expressions which involve the square and other powers of the trigonometrical ratios, such as (sin A)2, (tan A)3,... It is usual to write these in the shorter forms sin2 4, tan3 A,...

Example. Shew that sin2 A sec A cot2 A=cos A.

25. Το prove that sin2 A+ cos2 A = 1.

Let BAC be any acute angle; draw BC perpendicular to

a

b

AC, and denote the sides of the right-angled triangle ABC by

a, b, c.

Example 1. Prove that cos1 A - sin1 A = cos2 A - sin3 A.

cos1 A - sin1 A = (cos2 A + sin2 A) (cos2 A – sin2 A)

=cos2 4 - sin2 A,

since the first factor is equal to 1.

Example 2. Prove that cot a 1 - cos2 a = cos a.

COS a x sin a = cos a.