CHAPTER XVI.

ON THE RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE.

231. The three sides and the three angles of any triangle, are called its six parts.

By the letters A, B, C we shall indicate

geometrically, the three angular points of the triangle ABC; algebraically, the three angles at those angular points respectively.

By the letters a, b, c we shall indicate the measures of the sides BC, CA, AB opposite the angles A, B, C respectively.

232. I. We know that, A+B+C=180°. [Euc. 1. 32.] 233. Also if A be an angle of a triangle, then A may have any value between 0° and 180o. Hence,

(i) sin A must be positive (and less than 1),

(ii) cos A may be positive or negative (but must be numerically less than 1),

(iii) tan A may have any value whatever, positive or negative.

234.

(i)

Also, if we are given the value of

sin A, there are two angles, each less than 180", which have the given positive value for their sine.

(ii) cos A, or (iii) tan A, then there is only one value of A, which value can be found from the Tables.

235.

A B C
2+2 2

A

+ 90°. Therefore is less than 90°,

2

and its Trigonometrical Ratios are all positive.

A

Also, is 2

known, when the value of any one of its Ratios is given. Similar remarks of course apply to the angles B and C. Example 1. To prove sin (A+B)=sin C.

A+B+C=180° .. A+B=180o – C,

and .. sin (A+B)=sin (180o — C')=sin C.

[p. 104.]

2'

[Art. 118.]

Find A from each of the six following equations, A being an

Prove the following statements, A, B, C being the angles of a

A

2

B

C

sin34-sin+sin-1-2 cos 4. sin. cos

2

COS

(22) cos? 4

B

C

+ cos2

cos2

=

2 cos

2

2

2

(25) sin 24+ sin 2B+ sin 2C=4 sin A. sin B. sin C.

(26) sin A. cos A - sin B. cos B+sin C. cos C

2

=2 cos A. sin B. cos C.

=4 cos A. sin B. cos C.

(27) sin (B+C - A) − sin (C+A - B) + sin (A+B − C)

(28) cos 24+ cos 2B+cos 2C=-1-4 cos A. cos B. cos C. (29) sin2 A-sin2 B+sin2 C-2 sin A. cos B. sin C.

(30) cos (B+C−A)+cos (C+A − B) – cos (A+B-C)+1

(32) tan A+tan B+tan C÷tan A. tan B. tan C.

(33) tan. tan+tan. tan+tan 4. tan-1.

236. If ABC be a right-angled triangle having C = 90°, then (A + B) = 90o,

In a right-angled triangle ABC, in which C is a right angle, prove the following statements.

(13) sin(A-B)+cos 24=0. (14) sin (A-B)+sin(24+C)=0.

(15) (sin A-sin B)2+(cos A+ cos B)2=2.