2. Express cos 4 x in terms of sin x and cos x. cos 4 x = 2 cos2 2 x 1= 2(2 cos2 x-1)2 – 1 3. Deduce sin 30°, cos 30°, from cos 60°.

= 8 cost x

8 cos2x + 1.

4. Deduce sin 75°, cos 75°, from cos 150°. [Logarithms may be helpful.]

5. Deduce sin 221o, from cos 45°.

6. Deduce sin 15°, cos 15°, from cos 30°.

7. Express cos 6 x, sin 6 x, in terms of ratios of 3 x.

8. Express cos 3x, sin 3x, in terms of ratios of § x.
9. Express sin x, cos x, in terms of ratios of x.
10. Express cos 6 x, sin 6 x, in terms of ratios of 12 x.
11. Express cos 3 x, sin 3 x, in terms of ratios of 6 x.
12. Express sin x, cos x, in terms of ratios of 1⁄2 x.
13. Show that sin (n + 1)A + sin (n − 1) A = 2 sin nA cos A, and
cos (n + 1) + cos (n − 1) A = 2 cos nA cos A; and

hence, express sin 2 A, cos 2 A, in terms of sin A, cos A.

51. Tangents of the sum, and difference of two angles, and of twice an angle. Let A, B, be any two angles. It is required to find tan (A+B) and tan (A – B).

tan (A+B) =

sin A cos B+ cos A sin B

=

sin (A + B)
cos (A+B) cos A cos B- sin A sin B

On dividing each term of the numerator and the denominator of the second member by cos A cos B, there is obtained

Formula (2) can also be deduced from (1) by changing B into

Formulas (1), (2), (3), can be translated into words, as follows:

1. tan P= 2, tan Q = 3. Find tan (P + Q), tan (P – Q).

52. Sums and differences of sines and cosines. The set of formulas (1)–(4), Art. 50, can be transformed into two other sets which are very useful. From (1), (2), (3), (4), Art. 50, on addition and subtraction, there is obtained:

sin (A + B) + sin (A − B) = 2 sin A cos B.
sin (A + B) − sin (A – B) = 2 cos A sin B.

(1)

(2)

Substitution of these values of A, B, in (1)–(4) gives

Formulas (1)–(4) with the members transposed, are useful for transforming products of sines and cosines into sums and differences; formulas (5) to (8) are useful for transforming sums and differences of sines and cosines into products. These formulas may be translated into words: - Of any two angles,

2 cos one

cos the other

(3')

cos sum,

(41)

cos sum + cos difference,

2 sin one sin the other = cos difference

(71)

the sum of two cosines = 2 cos half sum · cos half difference, the difference of two cosines

2 sin half sum sin half difference, (8')

The difference between the first members of A, Art. 50, and C should be noted.

N.B. Arts. 92–95 are similar in character to, and are merely a continuation of, Arts. 50-52. If deemed advisable, Arts. 91-95 can be taken up now. The student is advised to glance at them after solving the following exercises:

3. Show that 2 sin (A + 45°) sin (A – 45°) = sin2 A — cos2 A.

2 sin (4+45°) sin (A–45°) = cos (A+45° — A−45°) — −cos (A+45°+A−45°),

-

Art. 52, B (4')

2 sin2 2 A − (1 − 2 sin2 3 A)] = sin2 3 A − sin2 2 A.

cos (xy), sin 2x, sin 2 y, cos 2 x, cos 2y, tan 2x, tan 2y, tan (x + y), tan (xy), when (a) both x, y, are in the first quadrant; (b) x is in the first, y in the second; (c) x in the second, y in the first; (d) both in the second quadrant. Check results by means of the tables.

29. Given sin x, sin y = 4.

3, sin y=- 2.

Do as in Ex. 28.

30. Given sin x = Find the ratios named in Ex. 28, when x is in the first quadrant and y in the third, x in the first and y in the fourth, x in the second and y in the third, x in the second and y in the fourth.

N.B. Examples suitable for exercise and review on the subject-matter of this chapter will be found in Arts. 91–95, and at page 187.