Because
radian measure is the ratio of two lengths, it is a unitless measure. For example, in
[link] , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.
Considering the most basic case, the
unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference,
$\text{\hspace{0.17em}}C=2\pi r,$ and for the unit circle
$\text{\hspace{0.17em}}C=2\pi .\text{\hspace{0.17em}}$ These two different ways to rotate around a circle give us a way to convert from degrees to radians.
In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in
[link] . Memorizing these angles will be very useful as we study the properties associated with angles.
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in
[link] , which are shown in
[link] . Be sure you can verify each of these measures.
Finding a radian measure
Find the radian measure of one-third of a full rotation.
For any circle, the arc length along such a rotation would be one-third of the circumference. We know that
Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.
$$\frac{\theta}{180}=\frac{{\theta}^{R}}{\pi}$$
This proportion shows that the measure of angle
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in degrees divided by 180 equals the measure of angle
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in radians divided by
$\text{\hspace{0.17em}}\pi .\hspace{0.17em}$ Or, phrased another way, degrees is to 180 as radians is to
$\text{\hspace{0.17em}}\pi .$
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
100•3=300
300=50•2^x
6=2^x
x=log_2(6)
=2.5849625
so, 300=50•2^2.5849625
and, so,
the # of bacteria will double every (100•2.5849625) =
258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.