A radian is a unit of measurement of an angle that can be defined as the measure of an angle that subtends an arc with a length of exactly one radius. This definition will be explained in more detail below.
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What do these terms have to do with measuring an angle? Imagine in that same circle if the radius {eq}\overline{AB} {/eq} is taken and laid it along the circle.
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The corresponding angle has a measure of exactly one radian, regardless of the radius of the circle.
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Measuring angles in radians can be helpful in finding arc lengths of a circle. The formula for arc length is {eq}l=r\theta {/eq} where l is the arc length, r is the radius of the circle, and {eq}\theta {/eq} is the angle subtended by the arc.
For example, the arc length of the following circle can be found using this formula. The radius of the circle is 10 inches, and the angle subtended by the arc is 0.7 radians. Therefore, the length of the arc is {eq}l = r\theta = (10\, \mathrm{in})(0.7 \,\mathrm{ rad}) = 7 \,\mathrm{ in} {/eq}.
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A circle has {eq}2\pi {/eq} radians or 360°. Similarly, the angle measure of a straight line is {eq}\pi {/eq} radians or 180°. Using this equivalency, conversion between degrees and radians can be done using the following formula: {eq}\frac{\mathrm{rad}}{\pi}=\frac{\mathrm{deg}}{180} {/eq} where rad is the angle in radians and deg is the angle in degrees. This formula can be manipulated to get a simple formula for converting an angle in degrees to one in radians: {eq}\mathrm{ rad}=(\pi)\frac{\mathrm{deg}}{180} {/eq} and for converting an angle in radians to one in degrees: {eq}\mathrm{ deg}=(180)\frac{\mathrm{rad}}{\pi} {/eq}.
The following examples will go through using the radian measure formula and converting between radians and degrees.
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In the above picture, angle {eq}\alpha {/eq} subtends an arc of length 1.5708 in a circle with radius 2. What is the measure of angle {eq}\alpha {/eq}?
The radian measure formula ({eq}l=r\theta {/eq} can be used to solve this problem. This formula can be rearranged to {eq}\theta=\frac{l}{r} {/eq} in order to solve for the angle {eq}\theta {/eq}. It is given that {eq}r = 2 {/eq} and {eq}l=1.5708 {/eq}. The unknown angle is {eq}\alpha {/eq}. Plugging in to the formula yields {eq}\alpha=\frac{1.5708}{2}=0.7854 {/eq}. Therefore, the measure of angle {eq}\alpha {/eq} is 0.7854 radians.
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In this example, the angle and arc length are given. This time, the radian measure formula can be used to calculate the radius of the circle. Again, the radian measure formula {eq}l=r\theta {/eq} can be rearranged to {eq}r=\frac{l}{\theta} {/eq}. The arc length is {eq}l=5 {/eq} and the angle subtending the arc has a measure of 2 radians. Plugging into the formula gives {eq}r=\frac{5}{2}=2.5 {/eq}. Therefore, the radius of the circle is 2.5 units.
In order to convert 89° into radians, the conversion formula is used: {eq}\mathrm{rad}=(\pi)\frac{\mathrm{deg}}{180} {/eq}. Plugging in 89° gives {eq}\mathrm{rad}=(\pi)\frac{89}{180}=1.55 {/eq}.
Therefore, 89° is approximately 1.55 radians.
A natural question that often arises upon learning about radians is why? Why would a mathematician use radians rather than the more familiar degrees?
Both radians and degrees have their place in math, as well as engineering, physics, chemistry, and more. Despite degrees being a more common form of measurement, radians are arguably much more intuitive. Pi is a fundamental mathematical concept, and it is fitting that a straight line has a measure of {eq}\pi {/eq} radians and that there are {eq}2\pi {/eq} radians in a circle. In fact, measuring angles in radians is where formulas like the circumference of a circle ({eq}2\pi r {/eq}) are derived from.
Aside from being intuitive, radians also make certain kinds of math simpler and more precise. As described above, radians are useful because the radian measure formula can only be used with angles measured in radians. Radians also make much of calculus easier, such as deriving trigonometric functions.
One radian is the measure of an angle that subtends an arc with a length of exactly one radian. Radians can be readily converted to and from degrees using the formula {eq}\frac{\mathrm{rad}}{\pi}=\frac{\mathrm{deg}}{180} {/eq}, where rad is the angle in radians and deg is the angle in degrees. The radian measure formula, {eq}l=r\theta {/eq}, describes the relationship between arc length, radius of the circle, and the angle that subtends the arc. In this formula, {eq}l {/eq} is the length of the arc, {eq}r {/eq} is the radius of the circle, and {eq}\theta {/eq} is the angle in degrees.
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There are 2π radians in a circle. Since one radian subtends an arc with a length of r, the circle's radius, this means that the circumference of the circle is 2πr.
A radian is an angle that subtends an arc of exactly one radius. One radian is equivalent to about 57.3°.
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