Maths Module 2

1.
In the question, it should be "...... the length of the ARC and that ofthe chord AB ....." instead.

Length of the arc AB
= 2 × π × 3 × (2.4/2π) cm
= 7.2 cm

Length of the chord AB
= 2 × [3 × sin (2.4/2)] cm
= 5.59 cm

OR : By cosine law, thelength of the chord AB
= √(3² + 3² -2 × 3 × 3 × cos2.4) cm
= 5.59 cm

Difference between the arc and that of the chord AB
= 7.2 - 5.59 cm
= 1.61 cm


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2.
Let r cm be the radius of the circle.

Perimeter of the sector :
r + r + 2 × π × r × (A/2π) = 14
2 × π × r × (A/2π) = 14 - 2r
π × r × (A/2π) = 7 - r ...... [1]

Area of the sector :
π × r² × (A/2π) = 10 ...... [2]

[2]/[1]：
r = 10/(7 - r)
7r - r² = 10
r² - 7r + 10 = 0
(r - 2)(r - 5) = 0
r = 2 or r = 5

Put the value of r into [1] :
π × 2 × (A/2π) = 7 - 2 or π × 5 × (A/2π) = 7 - 5
A = 5 or A = 0.8

The central angle A is 5 rad (a majorsector) or 0.8 (a minor sector).


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3.
L.H.S.
= (cosA + sinA) / (cosA - sinA)
= (cosA + sinA)² / [(cosA - sinA)(cosA + sinA)]
= (cos²A + sin²A +2 sinA cosA) / (cos²A - sin²A)
= (1 + 2 sinA cosA) / cos2A
= (1 + sin2A) / cos2A
= (1 / cos2A) + (sin2A / cos2A)
= sec2A + tan2A
= R.H.S.

Hence, (cosA + sinA) / (cosA - sinA) = sec2A + tan2A

2015-03-02 02:02:04 補充：
2 × π × 3 × (2.4/2π) = 71.06115… ?

It should be :
2 × π × 3 × [2.4/(2π)]
= 2 × π × 3 × [1.2/π]
= 2 × 3 × 1.2
= 7.2

2015-03-02 02:06:16 補充：
also, 7 - 5 = 2… not 0.8…?

It should be :
π × 5 × [A/(2π)] = 7 - 5
5 × (A/2) = 2
2.5A = 2
A = 2/2.5
A = 0.8

2015-03-02 02:10:15 補充：
Why (cosA - sinA)(cosA + sinA)=cos²A - sin²A ?

Let u = cosA and v = sinA

Formula: (u - v)(u + v) = u² - v²
Then, (cosA - sinA)(cosA - sinA) = (cosA)² - (sinA)²
i.e. (cosA - sinA)(cosA - sinA) = cos²A - sin²A

2015-03-02 02:11:32 補充：
The last two lines should be :
Then, (cosA - sinA)(cosA + sinA) = (cosA)² - (sinA)²
i.e. (cosA - sinA)(cosA + sinA) = cos²A - sin²A