Need help with trigonometry

• Mar 12th 2006, 12:41 PM
Confuzzled?
Need help with trigonometry
1. A chord PQ of a circle centre O, radius 10cm, subtends an angle of $\pi$/4 at the centre of the circle. Find the length of the major arc PQ and the area of the minor segment cut off by the chord.

2. The lengths of the side of a triangle are 10cm, x cm and (x-2)cm. The side of length (x-2)cm is opposite an angle of 60 degrees. Find x.

3. The area of a sector of a circle, diameter 7cm is 18.375cm squared. Find, without using a calculator, the length of the arc of the sector.

• Mar 13th 2006, 04:57 AM
ticbol
Quote:

Originally Posted by Confuzzled?
1. A chord PQ of a circle centre O, radius 10cm, subtends an angle of $\pi$/4 at the centre of the circle. Find the length of the major arc PQ and the area of the minor segment cut off by the chord.

2. The lengths of the side of a triangle are 10cm, x cm and (x-2)cm. The side of length (x-2)cm is opposite an angle of 60 degrees. Find x.

3. The area of a sector of a circle, diameter 7cm is 18.375cm squared. Find, without using a calculator, the length of the arc of the sector.

1)
Circumference, C = 2pi(10) = 20pi cm
Area, A = pi(10)^2 = 100pi sq.cm.

The minor arc has a pi/4 central angle, so the major arc has a (2pi -pi/4) = 7pi/4 central angle.
By proportion,
(maj. arc)/(7pi/4) = (Circumference)/(2pi)
(maj. arc) = [(20pi)/(2pi)]*(7pi/4)
(maj. arc) = 70pi/4 = 17.5pi cm ----------answer.

(Sector of pi/4 central angle)/(pi/4) = (area of circle)/(2pi)
sector = [(100pi)/(2pi)]*(pi/4) = 50pi/4 = 12.5pi sq.cm ---------***

Using A = (1/2)bc*sinA for area of triangle,
Area of (Isosceles triangle of pi/4 central angle) =
= (1/2)(10)(10)sin(pi/4)
= 50(1/sqrt(2)) = 50(sqrt(2) /2)
= 25sqrt(2) sq.cm. -------------------***

Therefore,
Area of minor segment = (area of minor sector) minus (25sqrt(2)
= 12.5pi -25sqrt(2)

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2)
Using Law of Cosines,
(x-2)^2 = 10^2 +x^2 -2(10)(x)cos(60deg)
x^2 -4x +4 = 100 +x^2 -10x
-4x +10x = 100 -4
6x = 96
x = 96/6 = 16 cm -----------answer.

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3)
Circle of diameter 7 cm.
Radius = 7/2 = 3.5 cm

Sector of the circle, area = 18.375 sq.cm.

18.375 = (1/2)(3.5)(arc)
arc = 2(18.375)/3.5 = 36.75/3.5 = 73.5/7 = 10.5 cm ---------answer.
No use of calculator.

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Only one problem, very detailed solution.
Three problems, (very detailed solution) divided by 3.
• Mar 13th 2006, 05:09 PM
Jameson
Quote:

Originally Posted by ticbol
1)
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Only one problem, very detailed solution.
Three problems, (very detailed solution) divided by 3.

Haha. I like this philosophy. :)