Question

A circle with a radius of 3.9 cm is centered at the vertex of an angle.Suppose the angle has a measure of 175 ____degrees.What is the radian measure of this angle?____ radians   What is the length (in cm) of the arc subtended by the angle's rays along the circle?_____ cm   Suppose θ represents the varying degree measure of the angle. Write an expression that represents the length (in cm) of the arc subtended by the angle's rays along the circle. (Enter "theta" for θ.) ______cm

Answer 1

We can draw the following picture:

From the angle-arc relationships, since the vertex is at the center of the circle, then the arc is equal to 175 degrees.

In radians, 175 degrees is equivalent to

$175=175\cdot(\frac{\pi}{180})\text{rad}$

that is

$175=3.054\text{ rad}$

What is the radian measure of this angle? 3.054 radians

The arc-lengh S is given by

$s=r\cdot\theta$

where, r=3.9cm and theta is equal to 3.054 rad (which is 175 degrees but in this formula the number must be written in radians). By sustituting these value, we have

$\begin{gathered} s=(3.9)(3.054) \\ s=11.91\text{ cm} \end{gathered}$

What is the length (in cm) of the arc subtended by the angle's rays along the circle? 11.91 cm

Suppose θ represents the varying degree measure of the angle. Write an expression that represents the length (in cm) of the arc subtended by the angle's rays along the circle.

We wrote the formula above:

$s=r\cdot\theta$

where s is the arc-lenght, r is the radius and theta is the angle (in radians).