Question

For each of the following angles, assume that the terminal ray of the angle opens up in the counter-clockwise direction a.

a) A circle with a radius 8 cm long is centered at Angle A's vertex, and Angle A subtends an arc length of 88 cm along this circle.

-The subtended arc is how many times as long as the circle's radius?

- Therefore, the radian measure of Angle A is?


b) A circle with a radius 18 cm long is centered at Angle B's vertex, and Angle B subtends an arc length of 72 cm along this circle. What is the radian measure of Angle B?


c) A circle with a radius 3 inches long is centered at Angle C's vertex, and Angle C subtends an arc length of 1.1781 inches along this circle. What is the radian measure of Angle C?

Answer 1

a) The subtended arc is 11 times longer than the radius.

ii. Angle A, subtended by the arc is 11 rad.

b) Angle B, subtended by the arc is 4 rad.

c) Angle C, subtended by the arc is 0.39 rad.

Explanation:

a) From the question, the radius of the circle is 8 cm and length of the arc is 88cm.

length of an arc can be determined by;

length of an arc = (θ/2
`\pi`) × 2
`\pi`r

where: r is the radius and θ is the angle subtended by the arc in radians.

So that;

length of an arc = θr

88 = 8θ

⇒ θ = 11

∴ length of an arc = 11r

The subtended arc is 11 times longer than the radius.

ii. Angle A, subtended by an arc, θ=
`(length of the arc)/(radius)`

⇒ θ =
`(s)/(r)`

=
`(88)/(8)`

= 11 rad

Angle A, subtended by the arc is 11 rad.

b) Angle B, subtended by an arc =
`(length of the arc)/(radius)`

⇒ θ =
`(s)/(r)`

=
`(72)/(18)`

= 4 rad

Angle B, subtended by the arc is 4 rad.

c) Angle C, subtended by an arc =
`(length of the arc)/(radius)`

⇒ θ =
`(s)/(r)`

=
`(1.1781)/(3)`

= 0.39 rad

Angle C, subtended by the arc is 0.39 rad.

Answer 2

a) 11 θ = 11

b) 4 θ = 4

c) 0.3927 θ = 0.3927

Explanation:

a) r = 8 cm length = 88 cm Then

The subtended arc is 11 times as long as the circle radius

the radian measure angle A is 11

b) r = 18 cm length = 72 cm

The subtended arc is 4 times as long as the circle radius

the radian measure angle A is 4

c) r = 3 in length = 1.1781 in

The subtended arc is 0.3927 times as long as the circle radius

the radian measure angle A is 0.3927