71. Double angles. To find the sine, cosine, tangent, and cotangent of twice a given angle in terms of the functions of the given angle.

From the formulas of Art. 68, letting ẞ= α, we have, after reduction,

To express clearly the real significance of these formulas, we may state them from two points of view.

If a be the angle under consideration, the formula

sin 2 α = 2 sin a cos a

may be stated: The sine of twice an angle is equal to twice the sine of the angle times the cosine of the angle.

If 2 a be the angle under consideration, the same formula may be stated: The sine of an angle is equal to twice the sine of half the angle times the cosine of half the angle.

It then follows that

sin a= 2 sina cosa.

(7)

Similarly, if a be the angle under consideration, the formula cos 2 α= cos2 α- sin2 a

may be stated: The cosine of twice an angle is equal to the square of the cosine of the angle minus the square of the sine of the angle.

If 2 a be the angle under consideration, the same formula may be stated: The cosine of an angle is equal to the square of the cosine of half the angle minus the square of the sine of half the angle.

72. Half angles. To find the sine, cosine, tangent, and cotangent of one half a given angle in terms of functions of the given angle.

From formula (9), Art. 71, we have

cos α = 1

α

2 sin2

In each of these formulas the positive or negative sign

α

is chosen to agree with the sign of the function of 2, depending on the quadrant in which lies.

α

2

These formulas may be considered from two view points. For example, formula (1) may be stated: The sine of half an angle is equal to the square root of the fraction whose numerator is one minus the cosine of the angle, and whose denominator is two; or, The sine of an angle is equal to the square root of the fraction whose numerator is one minus the cosine of twice the angle, and whose denominator is two.

73. To find the sum and difference of the sines of any two angles, also the sum and difference of the cosines of any two angles.

From the formulas of Arts. 68 and 69 we have, by addition and subtraction,

+ cos (a — ß) = 2 cos a cos ß

α

cos (a + B) cos (α — ẞ)= − 2 sin a sin ß.

Let a+ẞy and a-ẞd. Solving for a and ß, we have a (y+d) and ẞ=(y-3). Substituting these values in the above equation, it follows that

sin y + sin d = 2 sin (y+) cos

(1)

Equations (1), (2), (3), and (4) may be read from two view points.

Regarding and as the given angles, (1) may be stated: The sum of the sines of two angles is equal to twice the product of the sine of half the sum of the given angles into the cosine of half the difference of the given angles.

Thus,

sin 6x + sin 4 x = 2 sin 5 x cos x.

Regarding (y + ♂) and 1(y — ~) as the given angles, it is clear that their sum is y and their difference d. Then, by reading the second member first, equation (1) may be stated: Twice the sine of any angle times the cosine of any other angle is equal to the sine of the sum of the angles plus the sine of the difference of the angles.

Thus,

2 sin 20° cos 5° = sin 25° + sin 15°.

74. Equations and identities. The formulas developed in the present chapter are true for all values of the angles involved; hence they are trigonometric identities. By the use of these identities many others may be established. The remarks of Art. 54, concerning the use of the fundamental relations in establishing identities, apply here.

The identities of the present chapter are also useful in solving trigonometric equations. By their aid an equation involving functions of multiple angles may be transformed into an equation containing functions of a single angle (see Ex. 33, Art. 75). This transformed equation can then

may be much simplified by reducing sums or differences of sines and cosines to products by the relations of Art. 73 (see Ex. 41, Art. 75).

1. Find sin 221°, cos 221°, tan 2210, and cot 2210.

2. Given cos α = ; find sin 2a, cos 2α, tan 2α, and cot 2 α.

3. Given tan α=3; find sina, cosa, tanα, and cotα.

Prove the following identities: