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107. To find a general expression for all angles which have a given cosine.

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are angles with the same cosine, (Art. 101.)

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are angles with the same cosine. (Art. 105.)

Secondly, reckoning in a negative direction, - (2π-a) and -α

are angles with the same cosine.

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(1)

(2)

are angles with the same cosine, n being any positive integer.

Now the angles in (1) and (2) may be arranged thus:

2nπ+a, (2n+2) π − a, − (2n+2) π+a,

- 2nπ — α,

all of which, and no others, are included in the formula

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which is therefore the general expression for all angles which have a given cosine.

108. To find a general expression for all angles which have a

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are angles with the same tangent. (Art. 104.)

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are angles with the same tangent. (Art. 105.)

Secondly, reckoning in the negative direction,

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. (1)

(2)

are angles with the same tangent, n being any positive integer.

Now the angles in (1) and (2) may be arranged thus:

2nπ +α, (2n+1) π + a, − (2n + 2) π + α, − (2n + 1) π + a,

all of which, and no others, are included in the formula

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which is therefore the general expression for all angles which have a given tangent.

109. We shall now explain how to express the Trigonometrical Ratios of any angle in terms of the ratios of a positive angle less than a right angle.

First, when the given angle is positive.

If the angle is greater than 360o, subtract from it 360° or any multiple of 360°, and the ratios for the resulting angle are the same as for the original angle.

Thus we obtain an angle less than 360°, and if this angle be greater than 180°, we may subtract 180° from it, and the ratios for the resulting angle will be the same in magnitude, but the signs of all but the tangent and cotangent will be changed. (Art. 102.)

Thus we obtain an angle less than 180o, and if this angle be greater than 90°, we may replace it by its supplement, and the ratios for the resulting angle will be the same in magnitude, but the signs of all but the sine and cosecant will be changed. (Art. 102.)

Thus

sin 675o = sin (360° + 315°) = sin 315° = - sin 135°
tan 960o = tan (720° + 240°) = tan 240o = tan 60o.

Secondly, when the angle is negative.

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Add 360° or any multiple of 360° so as to obtain a positive angle, for which the ratios will be the same as for the original angle, and then proceed as before.

If the given angle be less than 180°, apply the formulæ obtained from Art. 100.

Ex. sin (-825°) sin (1080°-825°)= sin 255° - sin 75°,

=

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(3) sin 0:

EXAMPLES.-XXVI.

Write down the general value of which satisfies the follow

ing equations:

(1) sin 0 = 1.

=

1

√2

(2) cos 0=1.

(4) tan 0 = √3.

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110.

sec + 1 = 0.

The symbol sin1x denotes an angle whose sine is x,

(10)

sec2 0

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and a similar notation is used for the other ratios.

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CHAPTER X.

ON THE TRIGONOMETRICAL RATIOS OF TWO ANGLES.

111. We now proceed to explain the Trigonometrical Functions of the Sum and Difference of two angles. These functions are the most important in the subject, and the student will find that his subsequent progress will depend much on the way in which he has read this Chapter.

112. We shall first establish the following formulæ :

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

cos (A + B) = cos A. cos B-sin A. sin B,

sin (A – B) = sin A. cos B-cos A. sin B,

cos (AB) = cos A. cos B + sin A. sin B ;

by means of which we can express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the angles themselves.

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The diagrams which we shall employ are only applicable to the cases in which A and B are both positive and less than 90o, also, when we are considering the sum of the angles, A + B is less than 90°, and when we are considering the difference of the angles, A is greater than B. The formula are however true for all values of A and B. Particular cases may be proved by special constructions of the diagrams, but it is beyond the scope of this treatise to enter into detail on this and similar points.

8. T.

6

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