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173. We may also conversely prove that the 'A, B' formulæ are true for angles of any magnitude by the aid of the result of page 107. (This is a very convenient method of proving the ‘A, B' formulæ to be true for all values of the angles.)

For, assuming that the A, B' formulæ are true for certain values of the angles A and B, we can show that they are true if either of the angles A or B be increased by 90°.

Example. sin (90° + A+B)=COS (A+B)

[p. 107.] =cos A.. cos B - sin A. sin B, =sin (90° + A). cos B-{-cos (90° + A)} sin B,

=sin (90° + A). cos B+cos (90°+A). sin B. Or, writing A' for 90° + A, we have

sin (A' +B)=sin A'.cos B+cos A'. sin B.

We have proved (Art. 170) that the A, B' formulæ arə true for all values of A and B between 0° and 90°. And therefore, by what we have said above, they are true for all values of A or B between 0 and 180°. And so on.

Therefore the A, B’formulæ are true for any values whatever of the angles A and B.

174. It follows that all formulæ deduced from the · A, B' formulæ are true for angles of any magnitude what

ever.

Thus the 'S, T'' formulæ (page 126) are true for angles of any magnitude. Also the formulæ of the last Chapter for multiple angles, and all general formulæ in the Examples, are true for angles of any magnitude.

EXAMPLES, XLV.

(1) Deduce the values of sin 180° and cos 180° from those of the sine and cosine of 90°.

(2) The angle A is greater than 180° and less than 270o and tan A=i: find sin 2A and sin 3A.

(3) The angle 0 lies in the fourth quadrant and cos 6=1, find sin 20 and sin 38. Find also cos 38, and hence determine in which quadrant 30 lies.

(4) Prove that the different values of 0 which satisfy the equation cos po+cos q8=0, form two series in A. P. with com

21

27 mon differences

and respectively, p+9 p~9

ON SUBMULTIPLE ANGLES.

175. We have now proved all the formula of the last two Chapters to be universally true.

We may expect therefore that any result, which can be obtained from these formulæ by algebraical transformation, will have a complete geometrical interpretation. [See p. 116.]

A
176. Since,
cos A =1-2 sin?

[Art. 166.] 2

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Thus, given the value of cos A (nothing else being known

A about the angle A), we get two values for sin one posi

2'

A tive and one negative, and two like values for cos

2

177. To prove geometrically that, given the value of cos A (nothing else being known about the angle A), there

A

A are two values each of sin

and of cos 2

2

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Let a be the least positive angle which has the given cosine, and let ROP, and P,0R in the figure each = a. Then A is one of the angles described by the revolving line OP when, starting from OR, OP stops either in the position OP, or in the position OP,; i.e. any one of the angles

[Art. 147.)

2n = a.

a

Taking the halves of these angles in succession, we have the four dotted lines OP, OP, OP, OP, in the figure, as the lines indicating the various values of the

A angle ; i.e. any one of the angles

2• And it will be seen that

sin ROP, = sin ROP, =- sin ROPg=- sin ROPg. Also, cos ROPA cos ROPA = cos ROPg=cos ROP .

А From these it is clear that there are two values of sin

2' equal in magnitude and opposite in sign. Also, that there

А are two like values for cos

2

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EXAMPLES. XLVI.

(1) When A lies between - 180° and 180°, prove

that

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(2) When A lies between 180° and 540°, prove that

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(3) Find sin

2

in terms of cos A, when A lies between 180° and 3600

(4) Prove that when A lies between (4n+1) and (4n+3),

А

1+cos A n being a positive integer, cos

2 A (5) Find sin in terms of cos A, when A lies between Ant

2 and (4n+2) *, where n is a positive integer.

5

• COS

2

[Art. 166.]

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А А 178. Since, 2 sin

= sin A,

2

А A and,

sin?

+ cos? = 1,

2 2 we obtain by addition and subtraction А A

A А sin + cos + 2 sin

COS 1 + sin A, 2

2 2 A

А А sin?

2 sin COS 1- sin A.
2
2

2

A
Therefore

+ COS =l+sin A,
2

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A

+ cos?

+ cosa

sin

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1- sin A.

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(iii),

A
Adding we get, 2 sin Ni+sin A+ +V1-sin A

2

A Subtracting we get, 2 cos

=+11+sin AFVI-sin A 2

(iv).

Thus if we are given the value of sin A, (nothing else being known about the angle A), we have four values for A

A sin and four values for cos 2

s

2

179. To prove Geometrically that, given the value of sin A (nothing else being known about the angle A), there А

A are four values for sin and four values for cos 2'

2 Let a be the least positive angle which has the given sine, and let ROP, P,OL in the figure each = a.

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