Let 28 stand for a+b+c, so that s is half the sum of the sides of the triangle; then

Therefore

a+b−c=a+b+c−2c=2s−2c=2 (s—c),

a+c-b=a+b+c-2b=28-2b=2(s-b).

From the values of sin A and cos A we deduce

108. To express the sine of the angle of a triangle in terms of the sides.

Or we may proceed thus :

(sin 4)2 = 1 − (cos 4)2 = 1 − (3
- (b2 + co-a2)2

2bc

By comparing the two expressions for sin A we infer that

s(-a) (s-b) (s—c) =

2b2c2+2c2a2 + 2a2b2 — aa — ba — ca

16

and this can be verified by multiplying out the factors s, s-a, s-b, and s- c.

EXAMPLES. X.

1. If sin B =4, a=3, b=3, find A.

2. If A=75°, B=45o, b=2, shew that a=√3+1.

3. If b=c(3-1), and A-30°, find B and C.

4. If b=2a, and C=60°, find A, B, and c.

5. Find A when a=7, b=5, c=3.

6. If a, b, and c are 13 feet, 13 feet, and 2 feet respectively, find C.

7. The sides of a triangle are 7, 8, 13: find the greatest angle.

8. The sides of a triangle are respectively 13 and 15 feet, and the cosine of the included angle is 33: find the third side, and also the perpendicular on it from the given angle.

9. Shew from the formulæ for sin B and sin that C

B== if c2=b (b+a).

10. If a 5, b=4, and C=60°, find c; having given

log 45825=46611025, log 45826=46611120.

11. A perpendicular is drawn from the angle A of a triangle on the side BC meeting it at D; and a perpendicular from B on the side CA meeting it at E: shew that DE=c cos C.

12. Shew immediately from the figure in Art. 106 that

a=b cos C+c cos B.

13. From the result in the preceding example and the two analogous results deduce the value of cos C given in Art. 106.

14. If sin A=p/(1-q2)+q、√/(1 − p2), find cos A.

15. Shew that tan A

16.

=

sin A + sin 2A

1+cos A+ cos 24'

If tan A+ cot A = 2, then sin A+ cos A = √2.

17. Find A from the equation

sec A+ cosec2 = 3 sec1A.

18. The hypotenuse AB of a right-angled triangle is divided at D so that AD is to BD as CB is to CA: shew

19. From a ship at sea it is observed that the angle between two forts A and B is a; the ship sails for m miles towards A, and the angle between the forts is then observed to be ẞ; find the distance of the ship from B at the second observation.

XI. Solution of Triangles.

109. To solve a triangle having given two angles and a side.

Suppose A and C the given angles, and b the given side; then

therefore log a = log b + log sin A-log sin B

= log b + L sin A-10-L sin B+10

= log b + L sin A - L sin B.

Similarly log c=log b + L sin C− L sin B.

110. To solve a triangle having given two sides and the included angle.

Suppose b and c the given sides and A the included angle.