Question

Consider the angle shown below that has a radian measure of 2.9. A circle with a radius of 2.6 cm is centered at the angle's vertex, and the terminal point is shown.What is the terminal point's distance to the right of the center of the circle measured in radius lengths? ______radii   What is the terminal point's distance to the right of the center of the circle measured in cm?_______ cm   What is the terminal point's distance above the center of the circle measured in radius lengths?_____ radii   What is the terminal point's distance above the center of the circle measured in cm? _____cm   

Answer 1

Remember that we can use some trigonometric identities to find relations between distances in a circle when the central angle is provided:

If we measure each distance in radius lengths, it is equivalent to take r=1 on those formulas.

A)

The terminal point's distance to the right of the center of the circle, measured in radius lengths, would be:

$\cos (2.9\text{rad})=-0.9709581651\ldots$

This distance is signed since it indicates an orientation, but we can ignore the sign if we are only interested on the value of the distance.

Then, such distance would be approximately 0.97 radii,

B)

Multiply the distance measured in radius lengths by the length of the radius to find the distance measured in cm:

$0.97\times2.6cm=2.52\operatorname{cm}$

C)

The terminal point's distance above the center of the circle can be calculated using the sine function:

$\sin (2.9\text{rad})=0.2392493292\ldots$

Therefore, such distance is approximately 0.24 radii.

D)

Multiply the distance measured in radius length times the length of the radius to find the distance measured in cm:

$0.24\times2.6\operatorname{cm}=0.62\operatorname{cm}$