By Art. 148 any multiple of four right angles may be rejected. Thus we need only consider angles less than four right angles.

By Arts. 161 and 162 we can express the Trigonometrical Ratios of an angle between two and four right angles in terms of the Trigonometrical Ratios of an angle less than two right angles.

By Art. 95 we can express the Trigonometrical Ratios of an angle between one and two right angles in terms of the Trigonometrical Ratios of an angle less than a right angle.

Thus the statement is demonstrated.

164. It will be observed that when we thus reduce the angle we can always express a Trigonometrical Ratio of any angle in terms of the same Trigonometrical Ratio of the reduced angle.

165. As examples of the reduction of the angle wc have:

sin 700°=sin(360° + 340°) = sin 340° = sin (180° + 160°) =- sin 160°-sin 20°;

cos (-800°) = cos 800° = cos (720° +80°) = cos 80° ; tan 500°=tan (360°+140°)=tan 140o = -tan 40°;

cot 460° = cot (360°+100°)=cot 100o= — cot 80o.

sec 930° = sec (720° +210°) = sec 210° = sec (180°+30°) =-sec 30°;

cosec(-600°)=-cosec 600° cosec (360°+240°) = cosec 240° - cosec (180°+60°) = cosec 60°.

13. With the notation of Chapter XIV. shew that the area of a triangle is equal to

Rr (sin A+ sin B + sin C).

14. Shew also that the area is equal to

R2 (sin 2A + sin 2B+ sin 2C).

α,

15. From the bottom of a station in a horizontal plane the altitude of the summit of a mountain is found to be and on retiring c feet from the station its top is seen to be in a straight line with the top of the mountain: shew that if h be the height of the station the height of the mountain ch c-h cot a

is

feet.

16. If CD subtend an angle a at each of the stations A and B, which are distant h apart, and the sum of the two angles ABD and BAC be σ, shew that the distance between C and D is either

XVIII. Angles with given Trigonometrical Ratios.

166. We have shewn in Art. 17 that corresponding to a given angle there is only one value for an assigned Trigonometrical Ratio. But corresponding to a given value of an assigned Trigonometrical Ratio there is an unlimited number of angles, as we see from Chapter XV.

We shall now investigate expressions which include all the angles having a given value of an assigned Trigonometrical Ratio.

167. To find an expression for all the angles which have a given sine.

Let BOC be the least positive angle which has the given sine; denote this angle by A. Produce BO to any point B' and make the angle B'OC'=BOC; then BOC = 180° - A.

=

Now it is obvious from the figure that the only positive angles which have the same sine as A aro 180°-A, and the angles formed by adding any multiple of four right angles to A or to 180°-4; that is, angles included in the expressions n360° + A and n360°+180°-A, where n is zero or any positive integer. Also the only negative angles which have the same sine as A are-(180o +A) and — (360o — A), and the angles formed by adding to these any multiple of four right angles taken negatively; that is, angles included in the expressions n360°-(180°+A), and n360°-(360°- A), where n is zero or any negative integer.

All the angles which have been indicated will be found on trial to be included in the expression n 180o + ( − 1)" A, where n is zero or any integer positive or negative. Also

all the angles included in this expression will be found among the angles which have been indicated.

Thus this expression includes all the angles which have the same sine as A; and all the angles which it includes have the same sine as A. This expression also applies for all the angles which have the same cosecant as A.

168.

To find an expression for all the angles which have a given cosine.

Let BOC be the least positive angle which has the given cosine; denote this angle by A. Make the angle BOC'=BOĆ.

Now it is obvious from the figure that the only positive angles which have the same cosine as A are 360°-A, and the angles formed by adding any multiple of four right angles to A or to 360°-4; that is, angles included in the expressions n360°+ A and n360o +360o – A, where n is zero or any positive integer. Also the only negative angles which have the same cosine as A are -A and -(360o — A), and the angles formed by adding to these any multiple of four right angles taken negatively; that is, angles included in the expressions n360°-A, and n360o — (360o —A), where n is zero or any negative integer.

All the angles which have been indicated will be found on trial to be included in the expression n 360° A, where n is zero or any integer positive or negative. Also all the angles included in this expression will be found among the angles which have been indicated.

Thus this expression includes all the angles which have the same cosine as A; and all the angles which it includes have the same cosine as A. This expression also applies for all the angles which have the same secant as A.

169. To find an expression for all the angles which have a given tangent.

Let BOC be the least positive angle which has the given tangent; denote this angle by

any point B', and CO to any point 4. Produce BO to

C'.

Now it is obvious from the figure that the only positive angles which have the same tangent as A are 180°+A, and the angles formed by adding any multiple of four right angles to A or to 180°+ A; that is angles included in the expressions n 360°+A and n360°+180°+4, where n is zero or any positive integer. Also the only negative angles which have the same tangent as A are -(180°-4) and (360o-A), and the angles formed by adding to these any multiple of four right angles taken negatively; that is, angles included in the expressions n 360°-(180o-A) and n 360°-(360o-A), where n is zero or any negative integer.

All the angles which have been indicated will be found on trial to be included in the expression n 180°+A, where n is zero or any integer positive or negative. Also all the angles included in this expression will be found among the angles which have been indicated.

Thus this expression includes all the angles which have the same tangent as A; and all the angles which it includes have the same tangent as A. This expression also applies for all the angles which have the same cotangent as A.