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3. sin (x+y-z) = sin x cos y cos z + cos x sin y cos z

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4. sinx + siny - sinz - sin (x + y − z)

= 4 sin(x-2) sin(y — 2) sin(x+y). z)

5. sin (y − z) + sin (z − x) + sin (x − y)

+4 sin(y-z) sin (z — x) sin (x − y) = 0.

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49. Functions of Double Angles. To express the trigonometric functions of the angle 2x in terms of those of the angle x.

Put y=x in (1) of Art. 42, and it becomes

sin 2x sin x cos x + cos x sinx,

or

or

sin 2x2 sin x cos x

Put y=x in (2) of Art. 42, and it becomes

cos 2x cos2 x sin2x.

=1-2 sin2x.

=2 cos2x-1.

Put y=x in (1) and (3) of Art. 47, and they become

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Transposing 1 in (4), and dividing it into (1), we have

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(1)

(2)

(3)

(4)

(5)

(6)

(7)

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x is any angle, and therefore these formulæ will be true whatever we put for x.

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50. To Express the Functions of 3x in Terms of the Functions of x.

Put y = 2x in (1) of Art. 42, and it becomes

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= 2 sin x cos2 + (1 - 2 sin2 x) sinx (Art. 49)

x

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= (2 cos2x — 1) cos x - 2 sin2 x cos x (Art. 49)

=

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To express the func

tions of in terms of the functions of x.

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By formulæ (3), (4), and (5) the functions of half an angle may be found when the cosine of the whole angle is given.

52. If the Cosine of an Angle be given, the Sine and the Cosine of its Half are each Two-Valued.

By Art. 51, each value of cos x (nothing else being known about the angle x) gives two values each for sin and cos 2

one positive and one negative. But if the

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value of x be

given, we know the quadrant in which lies, and hence

we know which sign is to be taken.

х

2

Thus, if x lies between 0° and 360°, lies between 0° and

х

2

2

180°, and therefore sin sin is positive; but if x lies between 360° and 720°, lies between 180° and 360°, and hence

х

X 2

sin is negative. Also, if x lie between 0° and 180°, cos 20

2

is positive; but if x lie between 180° and 360°, cos is negative.

The case may be investigated geometrically thus :

Let OM the given cosine (radius being unity, Art. 16),

=

= cos x. Through M draw PQ per-
pendicular to OA; and draw OP, OQ.
Then all angles whose cosines are
equal to cos x are terminated either
by OP or OQ, and the halves of these A
angles are terminated by the dotted
lines Op, Oq, Or, or Os. The sines

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of angles ending at Op and Oq are

the same, and equal numerically to

B'

those of angles ending at Or and Os; but in the former case they are positive, and in the latter, negative; hence we

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obtain two, and only two, values of sin from a given value of cos x.

Also, the cosines of angles ending at Op and Os are the same, and have the positive sign. They are equal numerically to the cosines of the angles ending at Og and Or, but the latter are negative; hence we obtain two, and only two,

х

values of cos from a given value of cosæ.

2

Also, the tangent of half the angle whose cosine is given is two-valued. This follows immediately from (5) of Art. 51.

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