Page 39 - mathsvol1ch1to3ans

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8. The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having
the same radius. Express the angle of the sector in degrees, minutes and seconds.

Solution:Let the radius of a circle be R.

The perimeter of the semicircle will consist of the semicircular arc and the diameter= πR + 2R.
The perimeter of the sector is arc of the sector + 2R. Equating the two we get πR + 2R = arc of
the sector + 2R, or length arc of the sector = πR = R × θ in radians, or angle subtended by the arc
◦
at the centre is π radians or 180 .
9. An airplane propeller rotates 1000 times per minute. Find the number of degrees that a point on
the edge of the propeller will rotate in 1 second.
Solution:
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1000 rotations 1000 rotations 50 rotations 360 degrees 50 rotations
= = . Hence we have, × .
1 minute 60 seconds 3 seconds 1 rotations 3 seconds
360 × 50
Simplifying we get, degrees/second = 6000 degrees/second.
3
10. A train is moving on a circular track of 1500 m radius at the rate of 66 km/hr. What angle will it
turn in 20 seconds?
Solution:Speed of the train = 18.333 m per second.In 1 second, distanceis covered = 18.333m.

. In 10 seconds, distance is covered = 18.333 ∗ 10 = 183.333m.
Angle turned by the train moving on the circular curve in 10 seconds

= arc it made by moving on circular curve/radius of the circular curve

= 183.333/1500 = 0.1222

Hence, the train turnedby 0.1222 Radian (angular measurement)in 10 seconds which is equal to 7
degrees (approximately).
11. A circular metallic plate of radius 8 cm and thickness 6 mm is melted and moulded into a pie (a
sector of the circle with thickness) of radius 16 cm and thickness 4 mm. Find the angle of the
sector.
2
Solution: Volume of the metallic plate = πr h =π64(0.6) = π
Exercise 3.3:
1. Find the values of √
3 1 1
◦
◦
◦
(i) sin(480 ) = − (ii) sin(−1110 ) = − (iii) cos(300 ) =
2 2 2
1 1 √
◦
◦
(iv) tan(1050 ) = √ (v) cot(660 ) = −√ (vi) tan 19π = 3
√ 3 3 3
3

(vii) sin − 11π =
√ 3 2
!
5 2 6
2. , is a point on the terminal side of an angle θ in standard position. Determine the
7 7
trigonometric function values of angle θ.
√
v
! 2
u
2
u 5 2 6
p
2
2
r = x + y = t + = 1
7 7
√
!

y 2 6 x 5
sin θ = = ; cos θ = =
r 7 r 7
3. Find the values of other five trigonometric functions for the following: