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A

√(1+sin 4) (1),

and

A

sin 4-cos 4= √(1-sin 4) (2).

Thus as soon as the proper signs are known in (1) and (2) we can by addition and subtraction find expressions for

A

sin and cos

2

A
2

in terms of sin A.

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fore take the upper sign in (1) and the lower sign in (2), so

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204. As an example of the formulæ of the preceding Article we will find the sine and the cosine of an angle of 9o.

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2 cos 9o= √(1 + sin 18o) + √√(1—sin 18°);

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A

Thus if cos A be given and we require cos we have to

3

solve a cubic equation; and similarly if sin A be given and we require sin

A

we have to solve a cubic equation.

3

2

3. If sin 224°

EXAMPLES. XXI.

1. Find the cosine of 111o.

2. If A be between 90° and 180°, shew that

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√(1+sin A) + √√(1 − sin A).

-'69, write down the value of sin 112°.

A

4. Shew that tan

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double sign.

A

2

5. Shew that sin A when expressed in terms of sin has two equal values of opposite signs; and that cos A

when expressed in terms of cos

give a geometrical illustration.

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2

has only one value: and

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7. Shew that

sin 5A cosec2A sec A-cos 54 sec A cosec A=8 cot 24.

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sin 0+sin =m, cos 0+cos &=n, and cos (6+4)=P,

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XXII. Circular Measure.

206. In practice angles are always estimated by means of degrees, minutes and seconds; but there is another method of estimating angles which is very important in theory, which we will now explain. The object of the present Chapter is to establish and apply the following proposition: If with the point of intersection of any two straight lines as centre a circle be described with any radius, then the angle contained by the straight lines may be measured by the ratio which the length of the arc of the circle intercepted between the straight lines bears to the radius.

207. The circumferences of circles vary as their radii.

If a regular polygon of any number of sides be inscribed in a circle, and a regular polygon of the same number of sides be inscribed in another circle, the perimeters of the polygons are as the radii of the circles. See Art. 138. This is true however great be the number of the sides; and we may assume that by making the number of sides as large as we please the perimeters of the polygons will not differ sensibly from the perimeters of the corresponding circles.

For a fuller exhibition of this demonstration the student may consult Chapter II. of the larger work on Trigonometry.

208. Thus the ratio of the circumference of a circle to its radius is constant whatever be the magnitude of the circle; therefore of course the ratio of the circumference to the diameter is also constant. The numerical value of the ratio of the circumference of a circle to its diameter cannot be stated exactly; but it is shewn in the larger work on Trigonometry that the ratio may be calculated to

any degree of approximation. The value is approximately

355
113

equal to and still more nearly equal to ; the value

22
7

correct to eight places of decimals is 3.14159265...

The symbol is invariably used to denote the ratio of the circumference of a circle to its diameter; hence if r denote the radius of a circle its circumference is 2πr, where T=3.14159265...

209. The angle subtended at the centre of a circle by an arc which is equal in length to the radius is an invariable angle.

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With centre O and any radius OA describe a circle; let AB be an arc of this circle equal in length to the radius. Then, since angles at the centre of a circle are proportional to the arcs on which they stand,

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Thus the angle AOB is a certain fraction of four right angles, which is constant, whatever may be the radius of the circle.

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