LEAVING CERTIFICATE EXAMINATION.
1005
Value
isT
15.
15.
15
15
i5
15.
Alternative Questions.
4a. (1) If sin A = sin2 B, prove that
4 (cos 2A - cos 2B) = 1 - cos 45.
(2) Prove that
sec a + sec j3 + tan a - tan p
sec a + sec |3 - tan a + tan P
= tani (tt + 2a)cot |(v + 2/S).
Or
4&. Prove that
sin-1 (sin^ >/ 2) + sin~V cos 20 =|» all the angles being acute.
(2) If
a: = cos 0 + cos 0, y = sin 0 + sin <f>,
Prove that
1 cos J (0 + ^) = l sin i (0 + tp) =-a;a ^,"y?cos “ '/>)•
5a. State De Moivre’s theorem, and use it to prove that
„ . ,( n n(n-l)(n-2)(n-3)^ in )
os i?0=cosn 0 1 - ——tan2 0 + —^ y~~ 2'“ 3 4 ' tan ® 1
What is the corresponding expression for sin nQ 1
Use these expressions to find tan 60 in terms of tan 0
Or
56. Obtain the expansion of cos 0 in powers of 0, and write down
the corresponding expansion of sin 0.
Use these expansions to prove that
tan 0 = 0 + i 03,
where Q is so small that powers higher than the fourth may be
neglected, and hence find tan 10° correct to 4 places.
6a. Find the area of a quadrilateral in terms of its sides and of the
sum of two opposite angles.
From your result show that if the sides of a quadrilateral are given,
its area is greatest when it can be inscribed in a circle.
Or,
. A, B, C are 3 points in a straight line, such that
and P is a point outside the line, such that
2S.PBC\
nrnve that PA = PC ■
BC—3AB,
a.pcb=
9193
4 D