有幾題f.2數想請教下大家-)10points~~~

1) Prove the following identities:
a)
L.H.S.
= √(1 + tan2θ)
= √[1 + (sin2θ/cos2θ)]
= √[(cos2θ/cos2θ) + (sin2θ/cos2θ)]
= √[(cos2θ + sin2θ)/cos2θ]
= √(1/cos2θ)
= 1/cosθ
= R.H.S.


b)
L.H.S.
= 5 + 4sin(90o -θ) - sin2θ
= 5 + 4cosθ - (1 - cos2θ)
= 5 + 4cosθ - 1 + cos2θ
= cos2θ + 4cosθ + 4
= (cosθ)2 + 2(cosθ)(2) + (2)2
= (cosθ + 2)2
= R.H.S.


2)The age of a boy exceeds one-third of his father by 1. Five years later his age will be 4 less than half of his father's. What is the present age of his father?

Let f and b be the ages of the boy and his father respectively.

b - (1/3)f = 1 ...... (1)
(b + 5) - (f + 5)/2 = -4 ...... (2)

(1):
3[b - (1/3)f] = 3
3b - f = 3 ...... (3)

(2)
2[(b + 5) - (f + 5)/2] = -8
2(b + 5) - (f + 5) = -8
2b + 10 - f - 5 = -8
2b - f = -13 ...... (4)

(1) - (2):
b = 16

Put f = 16 into (3):
3(16) - f = 3
f = 45

The present age of his father = 45


3)Use Elimination

Case 1:
(6x - y - 4)/3 = 10 ...... (1)
(y + 3x - 7)/4 = 1 ...... (2)

(1):
6x - y - 4 = 30
6x - y = 34 ...... (3)

(2):
y + 3x - 7 = 4
3x + y = 11 ...... (4)

(3) + (4):
9x = 45
x = 5

Put x = 5 into (4):
3(5) + y = 11
y = -4

Hence, x = 5 and y = -4

Case 2:
6x - y - 4/3 = 10 ...... (1)
y + 3x - 7/4 = 1 ...... (2)

(1):
3(6x - y - 4/3) = 30
18x - 3y - 4 = 30
18x - 3y = 34
72x - 12y = 136 ...... (3)

(2):
4(y + 3x - 7/4) = 4
4y + 12x - 7 = 4
12x + 4y = 11
36x + 12y = 33 ...... (4)

(3) + (4):
108x = 169
x = 169/108

Put x = 169/108 into (4):
36(169/108) + 12y = 33
169/3 + 12y = 33
169 + 36y = 99
36y = -70
y = -70/36

Hence, x = 169/108 and y = -70/60