(i)

+

+

tan C

tan B tan C tan C tan A tan A tan B
=tan A+tan B+tan C − 2 (cot A+ cot B+cot C),
(ii) sin 4+ sin B+ sin C-1

and if A+B+С=nπ,

=4 sin (π- A) sin (π - B) sin † (π – C),

(iii) sin 24 + sin 2B + sin 2C = ± 4 sin A sin B sin C,
(iv) cos 24 + cos 2B + cos 2C+1:

4 cos A cos B cos C,

and determine when the upper sign is to be used.

x. Express the area of a triangle in terms of the sides.

Show how to construct the right-angled triangle of minimum area which has its vertices on three parallel lines; and if a, b are the distances of the middle line from the other two, show that the hypotenuse makes with the parallel lines an angle

If the given angle of the triangle instead of being a right angle is equal to a, find the angle which the side opposite to it makes with the parallel lines when the area is a minimum.

xi. Show how to solve a triangle when two sides and the included angle are given.

If two sides of a triangle are 71 and 25 feet and the contained angle 69° 32′ calculate the remaining angles and side, and show that if a small error has been made in the measurement of the smaller side it will affect the calculated value of the third side very slightly.

xii. Find the radii of the inscribed and escribed circles of a triangle; and if these are r, 71, 72, 73 and that of the circumscribed circle is R, prove that r1+2+T‍3¬r=4R.

If D, E, F are the centres of the escribed circles and O that of the inscribed circle, prove that

EF2 FD2 DE2 OD2 OE2 OF2

R

+
rr2 rr3

(5) sin A=%, cos A-, tan 4=§; sin B=, cos B=g, tan B=. (7) of the smaller angle, the sine, cosine=1, tangent=17. Of the larger angle, the sine=1, cosine, tangent=. (8) of the smaller angle, the sine, cosine= tangent= 9 2

√3

1

√3