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
Find the radian measure of three-fourths of a full rotation.
$$\frac{3\pi}{2}$$
Converting between radians and degrees
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
$\text{\hspace{0.17em}}\pi .$
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=