EXAMPLES. II.

Find the values of the other Trigonometrical Ratios in the following eight examples, having given :

Demonstrate the following identities :

9. (sin A+ cos A)2 + (sin A-cos A)2 = 2. 10. sin2 A-cos2 B = sin2 B-cos2 A. 11. sec2 A cosec2 A = sec2 A + cosec2 A. 12. sin1 A+ cos1 A = 1-2 sin2 A cos2 A. 13. tan A+cot A = secA cosec A. 14. sin A-cos1A = sin2A – cos2 A.

1-2 sin2 A cos2 A
sin A cos A

15. sin2 A tan A+ cos2 A cot A =
16. sin2 A+ vers2 A=2 (1-cos A).
17. sin3 A+ cos3 A = (sin A + cos A)(1 − sin A cos A).
18. sinR A+ cos6 A = sina A + cos1 A-sin2 A cos2 A.
19. sin2 A tan2 A+ cos2 A cot2 A = tan2A + cot2 A − 1.
20. sin A tan2 A + cosec A sec2 A-2 tan A sec A

2

21. (sin A cos B+ cos A sin B)2

=cosec A-sin A.

+(cos A cos B- sin A sin B)2 = 1.

22. (1+sin A+ cos A)2=2 (1 + sin A) (1 + cos A). 23. (1-sin A-cos A)2 (1 + sin À + cos A)2

= 4 sin2 A cos2 A.

24. (1+sin A−cos A)2 + (1 + cos A − sin A)2

=4(1-sin A cos A).

III. Values of the Trigonometrical Ratios for an angle of 45°, of 60o, of 30o.

26. In this Chapter we shall find the values of the Trigonometrical Ratios for certain angles.

27. To determine the values of the Trigonometrical Ratios for an angle of 45o.

Let BAC be an angle of 45°; take any point P in AC, and draw PM perpendicular to AB. Since PAM is half a right angle, APM is also half a right angle; therefore PM=AM.

28. To determine the values of the Trigonometrical Ratios for an angle of 60o and for an angle of 30°.

Let APB be an equilateral triangle, so that the angle PAB contains 60 degrees; draw PM perpendicular to AB, then AM=MB;

The Trigonometrical Ratios for an angle of 30o may be found by Art. 16: thus

29. The Trigonometrical Ratios for any angle can be found approximately; the ratios are seldom capable of being expressed exactly, as they are in the special cases which we have here considered, but the calculations may be carried to any assigned degree of accuracy. We shall not enter into an account of the processes of calculation in the present work, but may refer to the more complete treatise. It will be sufficient to state as a fact that tables may be easily procured which give to seven places of decimals the sine of any angle which can be expressed in degrees and minutes; the other ratios can be determined when the sine is known.

30. Although we shall not explain the mode in which the tables are constructed, yet the student will readily see as he proceeds with the subject that various formulæ occur which might be useful in calculating the values of the Trigonometrical Ratios. Especially he may notice the formulæ hereafter to be given by which we may determine the Trigonometrical Ratios for an angle which is the sum or the difference of two other angles having known Trigonometrical Ratios. And we shall give a formula in the next Article which will enable us to determine the Trigonometrical Ratios for the half of an angle when the Trigonometrical Ratios of the angle itself are known.

31. To express the tangent of half an angle in terms of the sine and cosine of the angle.

Let BOC be any angle, which we will denote by A.

Take any point P in OC. With O as centre, and OP as radius, describe a circle. Produce BO to meet the circumference at D. Draw PM perpendicular to OB; and join PD.

By Euclid III. 20 the angle PDM=ŁA.

32. By the preceding Article when sin A and cos A are known we can determine tan A; and then by Art. 24 we can deduce from tan 4 the values of the other trigonometrical ratios of 4.

For example, suppose A = 30°, then 4=15o.

we may multiply both numerator and denominator of the last fraction by 2-3, and thus we obtain the more convenient result tan 15°-2-3.

By Art. 23 sec2 15°=1+tan2 15=1+(2−√3)2=8-4/3. Hence we find sec 15° by taking the square root of 8-4√3; it is shewn in Algebra how to extract this square root; it is easy, by squaring both members, to verify that

8-4/3-(3-1)/2.