sin A (1+tan A) + cos A (1 + ctn A) = csc A+ sec A. 23. Prove

cos A+ cos (120° - A) + cos (120°+A)=0.

24. Prove

4 sin A sin (60° — A) sin (60° + A) = sin 3 A. 25. Given sin 3 0 + sin 2 0 + sin 0 = 0; find 0. 26. Given sin (x+α) = cos (x — α); find x. 27. Prove 2 sec 2 A= sec (45° +A) sec (45° — A).

In the next five formulas k denotes any integer. These formulas can be readily proved by making the proper substitutions in formulas 40, 41, 42, 43, and 20.

sin ka=2 cos (k—1) α sin a+ sin (k − 2) α.

cos ka- 2 sin (k-1) a sin a + cos (k—2) α.

33. Prove tank a =

tan (k 1) α + tan a

1

tan (k-1) « tan α.

34. Prove by the aid of the formula of Ex. 31 cos 3 A

4 cos3 A-3 cos A.

35. Prove that when A is between 0° and 90°, √1+sm A=1+2 sin A√1-sin A.

43. Determine the limits between which A must lie in order that

2 sin A++ sin 2 A-1- sin 2 A.

CHAPTER IV.

TRIGONOMETRIC TABLES.

§ 54. Functions of small angles.

Let AOв be any positive acute angle, and let a be its

Let AOB'-α, and therefore B'Oв: =2a; and let o be the centre of a circle of which the radius is rOA OB = OB′. Let the chord BB' cut оA at c; and let the tangents at B and B′ meet at T, which must lie in Oa produced. Then,

Now, we know by geometry, that

chord BB'< arc BB' < B'T+TB.

Hence, dividing through by 2r, and substituting the above values, we have,

sin a < a < tan α ;

[52]

or, the circular measure of an acute angle is greater than the sine of the angle, and less than the tangent of the angle.

The last written inequality may assume the form,

Now, multiplying through by sin a, we have —

1

But, if a = 0, cos a = sec a = 1; and the differences cos a (v. [53]) and sec a 1 (v. [54]) can be made less than any assigned quantity by taking a sufficiently small.

Hence,

where and

are positive quantities which decrease in

definitely when a decreases. We have then

sin α = (1 — ε) α, tana (1ε) α ; [56]

=

sin 1' tan 1′

0.0002908882....

§ 55. We have, by [30],

1 cos α= 2 sin2 α.

Now, by [55], –

=

sina (1") × } α = ↓ (1 — ε′′) α ;

... 1 — cos α = (1 — ¿′′)2 a 2,

1

or

or

[58]

cos α=1-(1 — ε′′) 2 a 2 > 1 — a2. [59]

If a is very small, since " decreases indefinitely with

§ 56. The formulas proved in the last two sections enable us to compute approximate values of the sine, cosine, and tangent of a very small angle; and the number of decimal places to which these values are correct depends on the smallness of the angle. Let us next seek the limits of error of these formulas, so that we may know in what cases we have a right to use them. Such limits may be found as follows:

Since sin α = 2 sina cosa, by [26], and since sina a cosa, by [53],

... sin aa cos 2α.

But since, by [52], sin α <α,

... cos2 « = 1 — sin2 a >1 — a 2;